· SERVIÇO DE PÓS-GRADUAÇÃO DO ICMC-USP Data de Depósito: Assinatura: ______________________...

UN

IVER

SID

AD

E D

E SÃ

O P

AULO

Inst

ituto

de

Ciên

cias

Mat

emát

icas

e d

e Co

mpu

taçã

o

Contact Anosov actions with smooth invariant bundles

Uirá Norberto Matos de AlmeidaTese de Doutorado do Programa de Pós-Graduação emMatemática (PPG-Mat)

SERVIÇO DE PÓS-GRADUAÇÃO DO ICMC-USP

Data de Depósito:

Assinatura: ______________________

Uirá Norberto Matos de Almeida

Contact Anosov actions with smooth invariant bundles

Doctoral dissertation submitted to the Institute ofMathematics and Computer Sciences – ICMC-USP,in partial fulfillment of the requirements for thedegree of the Doctorate Program in Mathematics.EXAMINATION BOARD PRESENTATION COPY

Concentration Area: Mathematics

Advisor: Prof. Dr. Carlos Alberto Maquera Apaza

USP – São CarlosFebruary 2018

Ficha catalográfica elaborada pela Biblioteca Prof. Achille Bassi e Seção Técnica de Informática, ICMC/USP,

com os dados inseridos pelo(a) autor(a)

Bibliotecários responsáveis pela estrutura de catalogação da publicação de acordo com a AACR2: Gláucia Maria Saia Cristianini - CRB - 8/4938 Juliana de Souza Moraes - CRB - 8/6176

d447cde Almeida, Uirá Norberto Matos Contact Anosov actions with smooth invariantbundles / Uirá Norberto Matos de Almeida;orientador Carlos Alberto Maquera Apaza. -- SãoCarlos, 2018. 163 p.

Tese (Doutorado - Programa de Pós-Graduação emMatemática) -- Instituto de Ciências Matemáticas ede Computação, Universidade de São Paulo, 2018.

1. Anosov Actions. 2. Contact Structures. 3.Geometric Structures. 4. Algebraic Action. I.Maquera Apaza, Carlos Alberto, orient. II. Título.

Uirá Norberto Matos de Almeida

Ações Anosov de contato com fibrados invariantes suaves

Tese apresentada ao Instituto de CiênciasMatemáticas e de Computação – ICMC-USP,como parte dos requisitos para obtenção do títulode Doutor em Ciências – Matemática. EXEMPLARDE DEFESA

Área de Concentração: Matemática

Orientador: Prof. Dr. Carlos Alberto Maquera Apaza

USP – São CarlosFevereiro de 2018

I dedicate this thesis to everyone who realised

that self-knowledge is a person’s truest possession and a most valued treasure.

ACKNOWLEDGEMENTS

Foremost, I thank CAPES and CNPq for financial support and fellowship.

I would like to express my gratitude to my advisor Prof. Carlos Maquera for thecontinuous support of my Ph.D study and research, for his patience, guidance, motivation andinsightful comments.

Besides my advisor, I would like to thank Prof. Thierry Barbot, who supervised meduring my stay in France, for his encouragement, hard questions and immense knowledge.

My sincere thanks to all my professors, who taught, guided and helped me during thispart of the never ending journey of learning. In particular I thank Prof. Carlos Grossi and Prof.Sasha Ananin who helped me to broaden my horizons with respect to mathematics and look atareas beyond my own.

I thank my family, for their continuous, unwavering and unconditional support, for theirlove and patience through the whole process of my PhD.

Last but not the least, I thank (in alphabetical order) Paulo Borges, Philipy Chiovetto,Felipe Dreilick, Rafael Esquines Ciro Ferreira Iago Israel, Christian Lemos, Ana Luchesi, MuriloLuiz, Guilherme Pereira, João dos Reis, and many others, dearest friends who supported me,both visibly in the most invisible ways, who made the darkest moments brighter and the brightones shine like the sun, who made my weights lighter and my joyful moments more plentiful,who cheered me up even when they didn’t know I needed cheering. I did not mention everybodyand every list will be undoubtedly incomplete and I apologise to all those people whose nameswere not included for one reason or another.

“As above, so below,

as within, so without,

as the universe, so the soul”

(Hermes Trismegistus)

“Acessamos o universo sob a lente de nossa perceção,

portanto as coisas que observamos no universo,

não podem ser mais que reflexos das estruturas internas de nossa mente”

(Carlos Grossi)

RESUMO

DE ALMEIDA, U. N. M. Ações Anosov de contato com fibrados invariantes suaves. 2018.163 p. Tese (Doutorado em Ciências – Matemática) – Instituto de Ciências Matemáticas e de Com-putação, Universidade de São Paulo, São Carlos – SP, 2018.

O problema da classificação dos sistemas Anosov são de grande interesse dentro da teoria dos sis-temas dinâmicos. Os principais exemplos conhecidos são de natureza algébrica e foi levantada nadécada de 1960 a conjectura de que estes são os únicos exemplos (SMALE, 1967). Esta conjec-tura se mostrou falsa para fluxos Anosov (ações de R), onde foram construídos contraexemplosem variedades de dimensões impares ((HANDEL; THURSTON, 1980) e (BARTHELMé et al.,)). Estes contra exemplos no entanto são de natureza patológica, e sob hipóteses um pouco maisfortes, por exemplo, suavidade dos fibrados invariantes, a conjectura permanece em aberto.

Em 1992, foi publicado um artigo (BENOIST; FOULON; LABOURIE, 1992) provando quefluxos de contato Anosov com fibrados invariantes suaves são de fato algébricos . Neste trabalhoprocuramos generalizar o resultado obtido em (BENOIST; FOULON; LABOURIE, 1992). Paraisso criamos uma definição adequada para ações de Rk contato Anosov, que generalizam a noçãode fluxo de contato Anosov, e seguindo a estratégia de prova utilizada em (BENOIST; FOULON;LABOURIE, 1992), obtivemos uma generalização parcial deste resultado.

Palavras-chave: Ações Anosov, Estruturas de Contato, Estrutura Geométrica, Ações algébricas.

ABSTRACT

DE ALMEIDA, U. N. M. Contact Anosov actions with smooth invariant bundles. 2018. 163p. Tese (Doutorado em Ciências – Matemática) – Instituto de Ciências Matemáticas e de Compu-tação, Universidade de São Paulo, São Carlos – SP, 2018.

The problem of classifying the Anosov systems is of great interest in the theory of dynamicalsystems. The most important known examples are of algebraic nature and it has been conjecturedon 1960’s by S. Smale (SMALE, 1967) that these are in fact the only examples. This conjecturehas been proved false for Anosov flows, where counter examples had been constructed for odddimensional manifolds ((HANDEL; THURSTON, 1980) and (BARTHELMé et al., )). Thisnon algebraic examples however are very pathological, and with some stronger hypothesis, forexample, smoothness of the invariant bundles, the conjecture remains open.

In 1992, it was published a paper (BENOIST; FOULON; LABOURIE, 1992) which proved thatcontact Anosov flows with smooth invariant bundles are in fact algebraic. In this monograph weseek to generalize the result obtained in (BENOIST; FOULON; LABOURIE, 1992). For thisend, we create an adequate definition for contact Anosov Rk-actions, and following the proofstrategy used in (BENOIST; FOULON; LABOURIE, 1992) we obtained a partial generalizationof this result.

Keywords: Anosov Actions, Contact Structures, Geometric Structures, Algebraic Actions.

CONTENTS

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 PRELIMINARIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Gromov’s Geometric Structures . . . . . . . . . . . . . . . . . . . . 22

2.2.0.1 Definitions and notations . . . . . . . . . . . . . . . . . . . . . . . 222.2.0.2 Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 Killing fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3.1 Canonical form on Fk(M) . . . . . . . . . . . . . . . . . . . . . . . 302.3.2 Lie Algebra of Killing Vector fields . . . . . . . . . . . . . . . . . 322.3.3 Killing Lie algebra and compatible connections . . . . . . . . . . 34

3 CONTACT ANOSOV ACTIONS . . . . . . . . . . . . . . . . . . . . 393.1 Anosov actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Generalized k-contact Structure . . . . . . . . . . . . . . . . . . . . 41

3.2.1 Generalized k-contact structures . . . . . . . . . . . . . . . . . . . 413.2.2 Other contact like structures . . . . . . . . . . . . . . . . . . . . . 48

3.2.2.1 Contact Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.2.2.2 r-Contact structures . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2.2.3 Multicontact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.2.2.4 Polycontact and Pluricontact . . . . . . . . . . . . . . . . . . . . . 54

3.3 Examples and Counter-examples . . . . . . . . . . . . . . . . . . . . 593.3.1 An example: SO(k,k+n)/SO(n) . . . . . . . . . . . . . . . . . . . 59

3.3.1.1 The action is Anosov . . . . . . . . . . . . . . . . . . . . . . . . . 613.3.1.2 The action is generalized k-contact . . . . . . . . . . . . . . . . . . 623.3.1.3 It is not k-contact . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.3.1.4 For k = 2 it is not a contact pair . . . . . . . . . . . . . . . . . . . 68

3.3.2 Geometric interpretation of SO(k,k+n)/SO(n) . . . . . . . . . . . 713.3.3 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.3.3.1 Anosovness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.3.3.2 Generalized k-contact . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.3.4 Algebraic actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4 MAIN THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.1 The geometric structure . . . . . . . . . . . . . . . . . . . . . . . . . 864.2 The model space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.2.1 An adapted connection . . . . . . . . . . . . . . . . . . . . . . . . 904.2.2 Bundle of volume forms . . . . . . . . . . . . . . . . . . . . . . . . 994.2.3 The Lie algebra g′ . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.2.3.1 Compatibility and the Isotropy group . . . . . . . . . . . . . . . . . 1054.2.3.2 g′ is reductive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.2.3.3 The semisimple part of the Levi decomposition . . . . . . . . . . . 112

4.2.3.3.1 Flatness of the determinant Bundle . . . . . . . . . . . . . . . . 1124.2.3.3.2 A semisimple Lie algebra . . . . . . . . . . . . . . . . . . . . . . 114

4.2.4 Building the Model Space . . . . . . . . . . . . . . . . . . . . . . 1174.3 Extending the Structure . . . . . . . . . . . . . . . . . . . . . . . . . 1234.4 Completeness of the structure . . . . . . . . . . . . . . . . . . . . . 127

5 GROUP REDUCTION . . . . . . . . . . . . . . . . . . . . . . . . . 1315.1 Parabolic subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 1325.2 Technical preparation . . . . . . . . . . . . . . . . . . . . . . . . . . 1385.3 Decomposing g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

6 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1476.1 Theorem 4.0.1 hypothesis . . . . . . . . . . . . . . . . . . . . . . . . 1476.2 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

APPENDIX A (G,X)-STRUCTURES . . . . . . . . . . . . . . . . . . 151A.0.1 Geometric structures . . . . . . . . . . . . . . . . . . . . . . . . . 151A.0.2 Developing map . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

APPENDIX B CONNECTIONS ON VECTOR BUNDLES . . . . . . 155B.0.1 Connections on Associated Vector Bundles . . . . . . . . . . . . 157

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

17

CHAPTER

1INTRODUCTION

The problem of classification is ever present in the world of mathematics, it has old roots,perhaps as old as mathematics itself, however, only after the turn of the 20th century, did it gainsuch a central role. A naive person may think it is just another expression of humanity’s propensityfor labelling things. However, in the fields of mathematics, the problem of classification has sucha robust and natural setting, that to call it merely a human propensity does not make it justice. Infew words, the problem of classification can be stated in the following way: Given an abstractobjet, and some of its properties, is there some "nice" list of families of examples such that thisobject belongs to one of those families?. ? In the field of Dynamical Systems, the problem ofclassification gave its first steps with the work of Henri Poincaré who realized that, given thedifficulty of solving, explicitly, a differential equation, the best one could hope was to give someinformation about the qualitative behaviour of the solutions and that this qualitative behaviourcan, sometimes, be enough to classify some dynamical systems.

It was some decades after Poincaré that the importance of hyperbolic dynamics wasrecognized. The seminal work of D. Anosov (ANOSOV., 1967), and latter on the works ofS. Smale (SMALE, 1967) shed some light on the importance of the hyperbolic behaviour forthe study of general dynamical systems. Since their introduction, the most important examplesof globally hyperbolic dynamics (Anosov Systems) have been algebraic in nature: toral auto-morphisms, their suspensions, and geodesic flows on (strictly) negatively curved manifolds,it was, therefore, only natural to conjecture (SMALE, 1967) that Anosov systems were, infact, algebraic, up to some (continuous) conjugation. This conjecture has been proved falsefor Anosov Flows, counter examples having been constructed by M. Handel and W. Thurston((HANDEL; THURSTON, 1980)) on dimension 3, and by T. Barthelme,C. Bonatti, A. Gogolevand F. R. Hertz ((BARTHELMé et al., )) on arbitrary odd dimensions. However the conjectureremains open for actions of higher rank abelian groups, but it can be shown that under someconditions Anosov flows and diffeomorphisms are in fact algebraic. For example, it was shownby J. Franks and S. Newhouse (FRANKS, 1970), (NEWHOUSE, 1970) that Zk Anosov actions

18 Chapter 1. Introduction

of co-dimension one, are topologically conjugated to hyperbolic toral automorphisms.

A related conjecture on the same direction was given by A. Verjovsky (VERJOVSKY,1974) that states that Anosov flows with co-dimension one on manifolds of dimension ≥ 4 aretopologically conjugate to a suspension of toral hyperbolic automorphism. This conjecture waslatter generalized to Anosov Rk actions on manifolds of dimension ≥ 3+ k. Some results in thisdirection were given by J. Plante (PLANTE, 1981), E. Ghys (GHYS., 1988) and T. Barbot andC. Maquera (BARBOT; MAQUERA, 2011).

Suspension of algebraic Anosov diffeomorphisms have the remarkable property that theinvariant bundles are jointly integrable, therefore, the exclude the rather important examplesof Anosov flows given by geodesic flows on negatively curved manifolds, for these flows areactually Reeb flows of a contact structure defined by the sub bundle E+⊕E− which is, therefore,nowhere integrable.

Taking into account this example, we cite another paper by E. Ghys who showed thatcontact Anosov flows on manifolds of dimension 3, with smooth invariant bundles, are in factgeodesic flows over surfaces of constant negative curvature. Latter on, in 1992, this result wasgeneralized by Y. Benoist, P. Foulon and F. Labourie (BFL - (BENOIST; FOULON; LABOURIE,1992)), who showed that smoothness of the invariant Anosov bundles implies that Anosovcontact flows are in fact algebraic (up to smooth conjugation, flow re-parametrization and finitecoverings). It is important to notice that the smoothness condition is in fact necessary, P. Foulonand B. Hasselblatt have shown ((FOULON; HASSELBLATT, 2013)) an example of contactAnosov flow which is not algebraic. It was BFL’s paper that motivated this work. We hoped toobtain a similar result for Anosov actions of Rk, confirming the standing conjecture about thealgebricity of higher rank abelian Anosov actions for some particular cases. We obtained partialsuccess:

Theorem 1.0.1. If M is a manifold which supports an Anosov action associated with a "gener-alized k-contact structure" with smooth invariant bundles, then it is smoothly conjugated to a"quasi-algebraic" action. If, moreover, the Anosov action is faithful and H1(π1(M),R) = 0 then,our "quasi-algebraic" model (G,K,Γ,a) can be take with G semisimple.

We also prove a kind of converse result

Theorem 1.0.2. Let G be a semisimple Lie group a a Cartan subspace with centralizer a⊕ k,and let K be the compact group associated with k. Let Γ be a uniform lattice on G acting freelyon G/K. Then there exists a generalized k-contact structure such that it’s associated action isprecisely the Weyl chamber action (that is, the right translation on Γ∖G/K by the exponentialsof a)

We also consider some cases with stronger hypothesis and prove the following tworesults:

19

Theorem 1.0.3. Under the hypothesis of Theorem 1.0.1, let (G,K,Γ,a) be our quasi-algebraicmodel. Suppose that G is an algebraic group. Then a contained in the Cartan subspace of G

Theorem 1.0.4. Under the hypothesis of Theorem 1.0.1, let (G,H,Γ,a) be our quasi-algebraicmodel. Suppose that H is a compact subgroup, then the action is the Weyl chamber action.

This work is organized in the following way:

Chapter 2: After we fix some notations on section 2.1, the sections 2.2 and 2.3 are devoted toreview some notions and results about Gromov’s geometric structures that will be of uselater on.

Chapter 3: The section 3.2 is devoted to defining our main object of study, Anosov actionsassociated with a generalized k-contact structure. On section 3.3 we construct a familyof examples prove Theorem 1.0.2. We also give a precise definition of algebraic actions(Definition 3.3.1) and quasi-algebraic actions (Definition 3.3.2).

Chapter 4: This chapter is devoted to the proof of first part of Theorem 1.0.1 which follows thefollowing structure

∙ Choose an adequate open subset Ω os M where a Lie pseudo-group acts transitively(section 4.1).

∙ Show that this open set is locally modelled after some homogeneous space G′/H ′

(section 4.2).

∙ Show that his local model can be extended to the whole manifold M and is complete(sections 4.3 and 4.4).

Chapter 5: This chapter is devoted to the proof of second part of Theorem 1.0.1, that is, to thestudy the Lie algebras g′ and h′ to obtain a reduction from G′ = G×Rk to G of the action(section 5). We also prove the Theorems 1.0.3 and 1.0.4

A large part of the strategy of the proof was outlined on BFL’s paper ((BENOIST;FOULON; LABOURIE, 1992)).

21

CHAPTER

2PRELIMINARIES

The main goals of this chapter are to fix our notations and to present some aspects of theGromov theory of geometric structures (Sections 2.2 and 2.3) which are crucial for the proof ofour main theorem.

2.1 Notations

In this chapter we stablish some notations.

∙ Plus or Minus (±) convention:

It is usual in mathematics to use the symbols ± or ∓ to denote the possibility to choseeither + or − on some expression. This may cause some ambiguity if the symbols ±,∓appear more the once in a single expression. For example, what could mean the expression:a±b± c∓d? Do we make a choice for each symbol ±,∓?

On this monograph, we adopt the convention that each expression allows for a singlechoice (as far as signs are concerned): top sign or bottom sign. For example the expressiona±b± c∓d can only have two possible meanings:

a+b+ c−d (top sign)

or

a−b− c+d (botton sign)

∙ Differential Manifolds

– We denote by Mm a smooth manifold of dimension m. If the dimension is eitherunimportant or already fixed, we omit the index m, and denote simply by M.

22 Chapter 2. Preliminaries

– If ω is any p-form p ≥ 2, and k ≥ 1 then we denote

(ω)k = ωk = ω ∧·· ·∧ω︸︷︷︸

k−times

– If ω is a p+ 1-differential form on M and X is a vector field on M, the interiorproduct iX ω is the p-form defined as

iX ω(Y1, . . . ,Yp) = ω(X ,Y1, . . . ,Yp)

– The Lie derivative of a tensor T along a vector field X is denoted by LX T .

∙ Vector bundles Consider π : E → M a smooth vector bundle over a smooth manifold M.

– If U ⊂ M is an open set, we denote by Γ(U,E) the space of local sections over U .

– If E is a subbundle of the tangent bundle, the elements of Γ(U,E) are called vectorfields. If we want to emphasize the fact that U = M, we call them local vector fields.

– If p ∈ M, we denote by Ep the fiber π−1(p).

– Let α ∈ Γ(U,E), then we denote αp = α(p).

∙ Lie groups and algebras

– Lie groups will be denoted by capital Latin letters: G, H, K, etc.

– Lie algebras will be denoted by lower-case Gothic Latin letters: g, h, k, etc.

– Let G be a Lie group, it’s Lie algebra will be denoted by Lie(G).

– Let (g, [ · · ] be a real Lie algebra, then gC will denote it’s complexification g⊕ ig

with induced Lie bracket [ · · ]C:

[u+ iv,x+ iy]C :=([u,x]− [v,y]

)+ i([v,x]+ [u,y]

)

2.2 Gromov’s Geometric Structures

In this section, we recall the basic facts about the theory of Gromov’s geometric structures.The theory is developed in a most general way, and follows closely the exposition by Benoist, Y.(BENOIST, 1997) and Feres, R. (FERES, 2002)

2.2.0.1 Definitions and notations

First, let fix some notations and conventions that shall be used on this chapter. Let M,N be manifolds of dimensions m and n respectively. We shall denote by Jk

p(M,N) the space of

2.2. Gromov’s Geometric Structures 23

k-jets1 from M to N on p. The elements of Jkp(M,N) will be denoted by either jk

p( f ) or jkf (p) or

jk( f )(p). We shall putJk(M,N) =

⊔p∈M

Jkp(M,N).

Both Jk(M,N) (and Jkp(M,N)) have a natural structure of smooth manifold. Let (U,x)

and (V,y) be local charts of M and N

Jk(U,V )−→ x(U)× y(V )×Lin(Rm,Rn)×k⊕

j=2

Sym j(m,n)

jkp( f ) ↦−→ (u,ϕ(u),ϕ ′(u), . . . ,ϕ(k)(u))

where u = x(p), ϕ = y ∘ f ∘ x−1, Lin(Rm,Rn) denote the space of linear maps from Rm to Rn

and Sym j(m,n) denotes the space of j-multi linear symmetric maps from Rm to Rn.

We shall also denote by Glk(m) ⊂ Jk0(R

m,Rm) the space of k-jets of local diffeomor-phisms from Rm to Rm that fixes the origin. A local map from M to N is a map f : U →V ⊂ M

defined on some open subset U of M. And for local diffeomorphisms we shall mean local mapsthat are diffeomorphisms. The composition law

jk0(ϕ) · jk

0(ψ) := jk0(ϕ ∘ψ)

makes Glk(m) a group. It is, in fact, a real, algebraic linear Lie group. We shall often identifyjk0(ϕ) ∈ Glk(m) with its Taylor polynomial’s coefficients: jk

0(ϕ)↔ (ϕ ′(0), . . . ,ϕ(k)(0)).

Lets also consider Fk(M) the space of k linear frames on M, that is the space of k−jetsof local diffeomorphisms from a neighbourhood of the origin in Rm to M. More precisely

Fk(M) := (p, jk0( f )) ∈ M× Jk

0(Rm,M) ; f : U0 →V ⊂ M,U0 open subset of Rm

f is a diffeomorphism such that f (0) = p

The canonical projection π : Fk(M)→ M defines a Glk(m)-principal bundle, where theaction of Glk(m) on Fk(M) is given by:

jk0( f ) · jk

0(ϕ) := jk0( f ∘ϕ) ∀ jk

0( f ) ∈ Fk(M) jk0(ϕ) ∈ Glk(m)

Definition 2.2.1. Let Σ be a smooth manifold, and let λ : Glk(m)×Σ → Σ be a smooth action.A geometric structure of order k and type λ (sometimes we say, of type Σ, if the action isunderstood), is a Glk(m)−equivariant map σ : Fk(M)→ Σ, that is

(x ·g) = λ (g−1,σ(x)) = g−1 ·σ(x)

1 A k-jet at p is a class of equivalent functions whose derivatives of order less then or equal to k coincideson some neighbourhood of p.


If Σ is a real smooth algebraic manifold and the action λ is algebraic, then σ is said to be anA-structure.

Remark 2.2.2. A geometric structure can also be thought as a section of the bundle Σ×Glk(m)

Fk(M)

Definition 2.2.3. An automorphism of σ is a diffeomorphism F of M, such that

σ ∘F(k) = σ

where

F(k) : Fk(M)→ Fk(M)

jk0(ϕ) ↦→ jk

0(F ∘ϕ).

If H : U →V is a local diffeomorphism, then the local automorphism is defined in thesame way with some obvious modifications:

H(k) : Fk(U)→ Fk(V )

jk0(ϕ) ↦→ jk

0(H ∘ϕ)

and we require that

σ|Fk(V )∘H(k) = σ|Fk(U)

The (pseudo) group of (local) automorphisms is denoted by Aut(σ)(Autloc(σ)).

Example 2.2.4. An example of especial interest for us is given by the pseudo-riemannian metrics.Let g be a pseudo-riemannian metric of signature s on M. We shall see how g can be understoodas an A-structure of order 1 on M. Let Σ denote the space of all symmetric bilinear forms ofsignature s on Rm. Clearly, Σ can be identified with the space GL(m,R)/O(s,m− s), whereO(p,q) denotes the isotropy group of the bilinear form

Q(x,x) = x21 + · · ·+ x2

p − x2p+1 −·· ·− x2

p+q

GL(m,R) acts on Σ by conjugation, that is, for a bilinear form B of signature s, andg ∈ GL(m,R) we have,

(g ·B)(u,v) := B(g−1u,g−1v).

Now, if we identify Gl1(m) with GL(m,R) (( j10(ϕ))↔ ϕ ′(0)), and we identify

F1(M) = (p,ξ ) ; p ∈ M , ξ : Rm → TpM is a linear isomorphism


then, we define the A-structure g of type Σ by

g : F1(M)→ Σ

(p,ξ ) ↦→ g(p,ξ )(u,v) := gp(ξ u,ξ v)

It is clear that the (local) automorphisms of g are in fact the (local) isometries of g.This motivates Gromov to call the automorphisms of a geometric structure, an isometry forthat geometric structure, but we shall reserve the name isometry for the automorphisms of thegeometric structure given by a Riemannian metric.

Like many things in differential geometry, geometric structures can be glued togetherfrom compatible local data. Let us make this notion more precise.

Definition 2.2.5. Let σ be a geometric structure of order k on M and let x : U ′ →U ⊂ Rm be alocal chart of M. We define the local form σx : U → Σ of σ with respect to x as

σx(u) := σ( jk0(x

−1 ∘ τu)) τu : h ↦→ h+u

Remark 2.2.6. If y : V ′ →V is another local chart, and ϕ := y∘x−1 has non empty domain, then

jk0(x

−1τu) = jk

0(y−1 ∘ϕ ∘ τu)

= jk0(y

−1 ∘ τϕ(u)) ·gkϕ(u)

where gkϕ(u) = jk

0(τ−ϕ(u) ∘ϕ ∘ τu) can be seen as the Taylor polynomial of ϕ at u.2.

Thus,

gkϕ(u)σx(u) = σy(ϕ(u)) (2.1)

Proposition 2.2.7. Consider a manifold Σ with a Glk(m) action. Suppose that, for a given atlasof M, for any coordinate system x : U →U ′ ⊂ Rm of this atlas we have a map σx : U → Σ suchthat, for any two coordinate systems x,y, with non zero intersection, we have,

gkϕ(u)σx(u) = σy(ϕ(u))

where ϕ := y ∘ x−1 and gkϕ(u) = jk

0(τ−ϕ(u) ∘ ϕ ∘ τu). Then, there exists a unique geometricstructure σ : Fk(M)→ Σ such that its local form is given by σx

Proof. Notice that if U ′ ⊂ Rm, is an open set then we have the identification Fk(U ′) = U ′×Glk(m). To see this, we consider, without loss of generality, U ′ to be an open ball around the

2 From this interpretation, it’s easy to see that gkϕ(u)

−1 = gkϕ−1(ϕ(u)) and gk

ϕ∘ψ(u) = gkϕ(ψ(u))gk

ψ(u)


origin and notice that any local diffeomorphism from U ′ to itself can be uniquely written as acomposition of a translation with a diffeomorphism that preserves the origin.

This identification allow us to identify the bundle Fk(U ′)×Glk(m) Σ with U ′×Σ. Themaps σx, therefore, allow us to build a section ξx of this bundle, and moreover, the coordinates x

allow us to view this as a subsection of the bundle Fk(U)×Glk(m) Σ.

It is a classical result that local, compatible smooth sections of a smooth bundle gluetogether on a global section. And the condition gk

ϕ(u)σx(u) = σy(ϕ(u)) does means, in fact, thatthe sections ξx are compatible.

2.2.0.2 Rigidity

Intuitively, rigidity (of order r) means that the r + 1-jets of automorphisms of somestructure, are determined by the r-jets of automorphisms. To make this notion precise, we shalldefine something called the prolongation of a geometric structure.

Definition 2.2.8. Consider Σ with a Glk(m) action and r = l + k, l ≥ 0. Define Σl := Jl0(Rm,Σ).

There exists a natural action λ l of Glr(m) on Σl given by:

λ l : Glr(m)×Σl −→ Σl

( jr0(ϕ, jl

0(()s))) ↦−→ jl

0((gk

ϕ · s)∘ϕ−1)= jl0(u ↦→ (gk

ϕ(ϕ−1(u))s(ϕ−1(u)))

)Moreover, the natural map Σl → Σ is Glr(m) equivariant.

A prolongation of degree l of a geometric structure σ is a geometric structure σ l whichmakes the following diagram commutative and equivariant

Fk+l(M) Σl

Fk(M) Σ

M

σ l

σ

πpk+lk

pk

Remark 2.2.9. We must verify that λl actually defines an action:


For jr0(ϕ), jr

0(ψ) ∈ Glr(m) and jl0(s)∈ Σl we have:

( jr0(ϕ) jr

0(ψ)) · jl0(s)= jr

0(ϕ ∘ψ) · jl0(s)= jl

0((gk

(ϕ∘ψ) · s)∘ (ϕ ∘ψ)−1)= jl

0((((gk

ϕ ∘ψ) ·gkψ) · s

)∘ (ϕ ∘ψ)−1)

= jl0(u ↦→ (gk

ϕ ∘ψ)(ϕ ∘ψ)−1(u) · (gkψ)(ϕ ∘ψ)−1(u) · s((ϕ ∘ψ)−1(u))

)= jl

0(u ↦→ gk

ϕ(ϕ−1(u)) · (gk

ψ)(ψ−1 ∘ϕ

−1(u)) · s(ψ−1 ∘ϕ−1(u))

)= jl

0((

gkϕ · ((gk

ψ ∘ψ−1) · s∘ψ

−1))∘ϕ

−1(u)))

= jr0(ϕ) ·

(jl0((gk

ψ ∘ψ−1) · s∘ψ

−1))= jr0(ϕ) ·

(jr0(ψ) · jl

0(s))

Proposition 2.2.10. For any l ≥ 0, there exists a natural l-prolongation of σ given by:

σ l : Fr(M) −→ Σl

jr0( f ) ↦−→ jl

0(u ↦→ σ( jk

0( f ∘ τu)))

Proof. We must verify that σ l is an Glr(m)-equivariant map, but before that, assuming thatσ l have been correctly defined. Then it is clear that the Diagram 2.2.8 is commutative andequivariant, that is both π and pk+l

k are Glr(m)-equivariant3.

Now, to prove that σ l is Glr(m)-equivariant, let jr0(ϕ) ∈ Glr(m) and jr

0( f ) ∈ Fr(M).From the definition of gk

ϕ , it follows that

jk0( f ∘ϕ ∘ τu) = jk

0( f ∘ τϕ(u)) ·gkϕ(u)

Also note that gkϕ(u) can be understood as an element of Glk(m). Thus, as σ is Glk(m)-

invariant

σl( jr

0( f ) · jr0(ϕ)) = σ

l( jr0( f ∘ϕ)) = jl

0(u ↦→ σ( jk

0( f ∘ϕ ∘ τu)))

= jl0(u ↦→ σ( jk

0( f ∘ τϕ(u)) ·gkϕ(u))

)= jl

0(u ↦→ gk

ϕ(u)−1 ·σ( jk

0( f ∘ τϕ(u))))

= jl0(u ↦→ gk

ϕ−1(ϕ(u)) ·σ( jk0( f ∘ τϕ(u)))

)= jl

0((

gkϕ−1 ·σ( jk

0( f ∘ τ∙)))∘ϕ)

)= λ

l( jr0(ϕ

−1),σ( jk0( f ∘ τ∙))

)= λ

l( jr0(ϕ)

−1,σ(v ↦→ jk0( f ∘ τv))

)= jr

0(ϕ)−1 ·σ l( jr

0( f ))

3 We understand that Glr(m) acts on Fk(M) and Σ via the natural map

Glr(m) ∋ jk+l0 (ϕ) ↦→ jk

0(ϕ) ∈ Glk(m)


Definition 2.2.11. Let Dk(M)⊂ Jk(M,M) be the open set of all k−jets of local diffeomorphismsof M. We define

Autkpq(σ) :=jkp(F) ∈ Dk(M); F(p) = q ;σ( jk

0(F ∘ f )) = σ( jk0( f )) ; ∀ jk

0( f ) ∈ Fkp (M)

From the Glk(m)-equivariance of σ it follows that we can substitute the "for all" on the abovedefinition with "for some". For r = k+ l, l ≥ 0 we also define

Autrpq(σ) := Autrpq(σl)

For convenience, we also put Autr(σ) =⊔

p,q∈M Autrpq(σ). The elements of Autr(σ) are calledinfinitesimal automorphisms of order r.

Definition 2.2.12. There is an obvious projection πrpq : Autr+1

pq (σ) → Autrpq(σ). A Gromovr-rigid geometric structure (or simply r-rigid geometric structure) is a geometric structure suchthat πr

pp is injective for every p ∈ M.

Lemma 2.2.13. Pseudo-riemannian metrics are 1-rigid.

Proof. Let’s first remember the proof of a well known result:

If ϕ,ψ : U → M are local isometries (in the traditional sense) then ϕ = ψ if, and only if,

there exists a point x in each connected component of U such that dϕx = dψx.

The proof is rather easy: we suppose that U is connected, and let V = y ∈U ;ϕ(y) =

ψ(y) and dϕy = dψy. It is clear that V is closed and the isometries ϕ and ψ coincide on V . Itremains to show that V is open. Let y ∈V and let W be a normal neighbourhood of y. Then, forevery z ∈W there exists v = v(z) ∈ TyM such that z = expy(v). This implies that

ϕ(z) = ϕ(expy(v(z))) = expϕ(y) dϕyv(z)

= expψ(y) dψyv(z) = ψ(expy(v(z))) = ψ(z)

and thus z ∈V .

Consider g : GL(m,R)→ Σ ≡ GL(m,R)/O(s,m− s) the geometric structure associatedwith a pseudo Riemannian metric, and put g = g1 be the 1-prolongation of g. We want to showthat π2

pp : Aut2pp(g) → Aut1pp(g) is injective. But j10(ϕ) ∈ Aut1pp(g)if, and only if ϕ is a local

isometry at p, and, as we have seen, the local isometry ϕ is completely determined by its firstderivative, which means that π2

pp is injective.

Two important results about geometric structure are the following:

2.3. Killing fields 29

Theorem 2.2.14 (Gromov, (GROMOV, 1988), Theorem 1.6.F). If σ is a rigid geometric structureof order r on M, there exists k and an open dense, Autr+k(σ)-invariant, subset M0 ⊂ M suchthat every infinitesimal automorphisms in Autr+k

v1,v2(σ), v1,v2 ∈ M0, extends to a unique local

automorphism. In other words, the map

Autlocv1,v2

(σ)→ Autr+kv1,v2

(σ)

f ↦→ jr+kv1

( f )

is injective for every v1,v2 ∈ M0.

Corollary 2.2.15. If σ is rigid, then there exists an open dense subset M0 such that, for suffi-ciently large r, the orbits (on M0) of Autloc(σ) and Autr(σ) coincide.

Theorem 2.2.16 (Gromov,(GROMOV, 1988),Theorem 3.1). If σ is an algebraic (possibly norigid) geometric structure of order r on M, then, for every k there exists a finite partition of M inlocally closed subsets

M = Mk0 ∪Mk

1 ∪·· ·∪Mksk

such that

1. Mk0 is an open dense subset in M and, for every 1 ≤ l ≤ sk, the union Mk

l ∪Mkl+1∪·· ·∪Mk

sk

is closed in M.

2. This partition is Autr+k(σ)-invariant.

3. The partition Mk+1 is a refinement of the partition Mk.

4. The Autr+k−1-orbits in Mk0 are closed smooth submanifolds.

As a consequence of those two theorems, we obtain

Theorem 2.2.17 (O¯

pen-Dense orbit theorem). [Gromov, (GROMOV, 1988), Corollary 3.3.A]Let σ : Fk(M)→ Σ be an A-structure that is rigid (for some r), then a dense orbit of Autloc(σ) isopen.

2.3 Killing fields

The goal of this chapter is to study the infinitesimal symmetries of a geometric structure,that is, the vector fields that integrates to automorphisms of the geometric structure. After a shorttechnical preparation in the first section, we prove, in the second section, that such vector fieldsforms a Lie algebra. In the third section, we explore an interesting consequence of the existenceof an affine connection which is "compatible" with a geometric structure.


2.3.1 Canonical form on Fk(M)

Let Pk = Fk(Rm). For k = 1 this bundle has no natural group structure, however, it doeshave a distinguished element e, that is, the k−jet of the identity at the origin o ∈ Rm. We shalldenote by Pk the tangent space TePk. Note that for k = 1 this is precisely the Lie algebra ofthe affine Lie group and for k = 0 this is in fact Rn itself. The canonical form on F k shall be adifferential 1-form which takes values on Pk−1.

Remark 2.3.1. For r > k the projection Pr → Pk induces a projection Pr →Pk. The kernel ofthe projection Pk →P0 is in fact the Lie algebra of the group Glk(m).

Let u = jk0( f ) ∈ Fk(M) and u′ the image of u on the projection Fk(M)(M)→ Fk−1(M).

The local diffeomorphism f induces a map around the distinguished element e

f : Pk−1 −→ Fk−1(M)

jk−10 (g) ↦−→ f ( jk−1

0 (g)) = jk−10 ( f ∘g)

Notice that the map f is actually a diffeomorphism.

The differential of f at the distinguished element (which can be identified with j1e( f ))

maps Pk−1 onto Tu′Fk−1(M). This linear map depends only on u and not on the choice of f , andthus, we shall denote it u.

Definition 2.3.2. Let Z be a tangent vector of Fk(M) at u, that is, Z ∈ TuFk(M). Let Z′ ∈Tu′Fk−1(M) be the image of Z under the projection Fk(M)→ Fk−1(M). The canonical form θk

on Fk(M) is a Pk−1 valued 1-form defined by

θk(Z) = u−1(Z′)

It is immediate from the definitions, that if r > k and if v is the image of u under theprojection Fr(M)→ Fk(M), the following diagram:

Pr Tu′Fr(M)

Pk Tv′Fk(M)

u

v

is commutative, where v′ is the image of v under the projection Fk(M)→ Fk−1(M). Thus wealso have the following commutative diagram


T (Fr(M)) Pr−1

T (Fk(M)) Pk−1

θr

θk

Lemma 2.3.3. Let M and M′ be two m dimensional manifolds, and f : M → M′ be a localdiffeomorphism. Then, f induces a map f(k) : Fk(M)→ Fk(M′). Let θk and θ ′

k be the canonicalforms on Fk(M) and Fk(M′) respectively. Then

f *(k)(θ′k) = θk

Proof. Notice that we have the following commutative diagram

Fk(M) Fk(M)

Fk−1(M) Fk−1(M)

f(k)

f(k−1)

Thus, if Z is a tangent vector of Fk(M) at u, then (( f(k))*Z)′ = ( f(k−1))*(Z′). Let w =

f(k)(u). Thus,

f *(k)(θ′k)(Z) = θ

′k(( f(k))*Z) = w−1(( f(k−1))*Z′)

on the other hand, if u = jk0(h), then v = jk

0( f ∘h), also, for jk−10 (g) ∈ Pk−1,

f ∘h( jk−10 (g)) = jk−1

0 ( f ∘h∘g) = f(k−1)( jk−10 (h∘g)) = f(k−1) ∘ h( jk−1

0 (g))

thus

w = j10( f ∘h) = j1

0( f(k−1) ∘ h) = ( f(k−1))*( j10(h)) = ( f(k−1))* ∘ u

and the result follows.

Corollary 2.3.4. Every local diffeomorphism of M lifted to Fk(M), leaves the canonical forminvariant.

There is a kind of converse of this lemma

Lemma 2.3.5. Let M and M′ be two m-dimensional manifolds, and let f : Fk(M)→ Fk(M′) bea fiber preserving transformation such that

f *(θ ′k) = θk

Then, f = f(k) for some local diffeomorphism f .


Proof. Because f is fiber preserving, it induces a transformation f from M to M′. We shallprove that f = f(k). Let Φ = f−1

(k) ∘ f . Then, Φ is a fiber preserving transformation that inducesthe identity transformation on M. Moreover, it leaves θk invariant, that is, if Z ∈ TuFk(M) thenθk(Z) = θk(J*Z). But, we notice, that the diagram 2.3.1 can be generalized for diffeomorphismsthat preserves the fiber, that is, if Φ is a local diffeomorphism on Fk(M) that preserves the fiber,then, it induces a local diffeomorphism φ on Fk−1(M), and we have

Fk(M) Fk(M)

Fk−1(M) Fk−1(M)

Φ

φ

Repeating the same considerations of the previous lemma, we have

u−1(Z′) = θk(Z) = θk(Φ*Z) = (φ*u)−1(Z′)

which means that u = φ*u. As u is an isomorphism, this means that φ* = Id. Which means thatΦ = Id, as we desired.

2.3.2 Lie Algebra of Killing Vector fields

Definition 2.3.6. Let σ : Fk(M)→ Σ be a geometric structure on M. A Killing field for σ is avector field X such that the diffeomorphism defined by the flow ϕt : M → M are automorphism,that is, ϕt ∈ Aut(σ).

Notice that if ϕt is the flow of a vector field X on M, then, it induces a one parameterfamily of diffeomorphisms (ϕt)(k) on Fk(M). The infinitesimal generator of this family will bedenoted by X(k) and is called the Natural Lift of X to Fk(M).

Lemma 2.3.7. A vector field X is a Killing field for σ if, and only if, dσ(X(k)) = 0

Proof. If X is a Killing field, then the flow ϕt ∈ Autloc(σ), that means that

σ((ϕt)(k)) = σ

But this means that (ϕt)(k) ∈ Autk(σ). Thus, dσ will collapse the vectors tangent to (ϕt)(k). Thatis dσ(X(k)) = 0.

Reversing the implications above, we obtain the other implication.


There is a good characterization of the vector fields on Fk(M) which are natural lifts, ituses the canonical form on Fk(M).

Theorem 2.3.8. For each vector field X of M, there exists a unique vector field X on Fk(M)

such that

1. X is invariant by the action of Glk(m).

2. LX θk = 0, where θk is the canonical form on Fk(M).

3. πXu = Xπ(u) for every u ∈ Fk(M)

And conversely, if X satisfies 1−2, then there exists unique vector field X on M satisfying3.

Proof. Let X = X(k) be the natural lift of X to Fk(M). By definition, X(k) satisfies the thirdproperty. Also, as the flow (ϕt)(k) commutes with the right action of Glk(m), it follows that X isGlk(m)-invariant. The second property follows from Corollary 2.3.4.

To prove the uniqueness, let Y be another lift of X to Fk(M). Let ψt be the flow generatedby Y . The first property means that ψt commutes with the action of Glk(m). This means thatwe can project it to ψt a one parameter group of diffeomorphisms on M. From Lemma 2.3.5,this means that ψt = (ψt)(k), and from the third property, this means that ψt = ϕt , and thus,ψt = (ϕt)k and therefore Y = X(k).

Conversely, if X is a vector field satisfying 1− 2, then, for each x ∈ M let u ∈ Fk(M)

such that x = π(u) and set Xx = π(Xu). From 1, Xx is independent of the choice of u and thus X

satisfies 3. The uniqueness is trivial.

Corollary 2.3.9. The space of Killing vector fields of σ on M is a Lie algebra.

Proof. From Theorem 2.3.8, we have

˜[X ,Y ] = [X ,Y ]

for any vector fields X ,Y on M. And thus,

dσ([X ,Y ](k)) = dσ([X(k),Y(k)])

From Lemma 2.3.7, if follows that if X and Y are Killing vector fields, then so is[X ,Y ].

Corollary 2.3.10. Let Jk(T M) be the space of k-jets of sections of T M. For every u ∈ Fk(M),π(u) = x ∈ M, the spaces TuFk(M) and Jk

x (T M) can be naturally identified.


Proof. Let X be a local vector field on M around x. Taking X(k) the natural lift of X , it is easy tosee that X(k)(u) depends only on jk

x(X). The uniqueness of the lift makes the map

Jkx (T M)→ TuFk(M)

injective. Analysing the dimensions, we conclude that this is a isomorphism.

In particular, this means, that every tangent vector of TuFk(M) can be extended to aprojectable vector field.

Lemma 2.3.11. Fix a point o ∈ M. Let h′ denote the space of germs of killing fields that vanishat o. Then, h′ is the Lie algebra of H ′ = Autloc

oo (σ).

Proof. We have not actually defined the space Autlocxy (σ). It is defined as the space of germs of

local automorphisms sending x to y.

From definition, a local vector field X defined on a neighbourhood U of o is a (local)Killing vector field if, and only if, its flow is a local automorphism: σ ∘ (ϕt)(k) = σ . Moreprecisely:

jkx(ϕt) ∈ Autkxϕt(x)(σ) ; ∀ x ∈U ; ∀ t ∈ (−ε,ε)

If X vanishes at o, we have

jko(ϕt) ∈ Autkoo(σ) ; ∀ t ∈ R

This means that h′ ⊂ Lie(H ′). To prove the opposite inclusion, we must show, that everyone parameter subgroup of H ′ is of the form jk

o(ϕt) for ϕt the flow of a Killing vector field thatvanishes at o.

Let γ(t)⊂ H ′ be a smooth one parameter subgroup. For each t, γ(t) is the germ of a localautomorphism at o, that fixes o. We take a representative γ(t) = ψt such that

ψt is a local diffeomorphism near o ∈ M and jko(ψt) ∈ Autkoo(σ) (2.2)

But ψt is a smooth one parameter family of local diffeomorphisms, which has infinitesimalgenerator Y , and (2.2) means precisely that Y is a Killing field that vanish at o, which finishesthe proof.

2.3.3 Killing Lie algebra and compatible connections

Now, let us consider a smooth connection ∇ on M. We denote by T and R the torsionand curvature tensors. We shall suppose that ∇T = 0. We define the derivation AX = LX −∇X .


As this derivation is zero on the function algebra C∞(M), it follows that it is induced by a(1,1)-tensor, and thus, it makes sense to calculate AX v for any tangent vector v of M.

Remember that this is also true for the torsion and curvature tensors, that is, for tangentvectors v1,v2 ∈ TpM, T (v1,v2) is a well defined element of TpM4, and likewise, R(v1,v2) is welldefined endomorphism of TpM.

We have:

R(X ,Y ) = [∇X ,∇Y ]−∇[X ,Y ] = [LX −AX ,∇Y ]−∇[X ,Y ]

= [LX ,∇Y ]−∇[X ,Y ]− [AX ,∇Y ]

Definition 2.3.12. Let M be a manifold with a connection. A vector field X on M is called aninfinitesimal affine transformation if the flow ϕt of X is affine, that is it preserves the connection.

The following theorem is classic:

Lemma 2.3.13 (Kobayashi, S and Nomizu, K. (KOBAYASHI; NOMIZU, 1963), Section VI.2Proposition 2.2). Let M be a manifold with a connection, then, X is an infinitesimal affinetransformation if, and only if,

[LX ,∇Y ] = ∇[X ,Y ] for every vector field Y on M

Corollary 2.3.14. If X is an infinitesimal affine transformation then,

R(X ,Y )Z = [AX ,∇Y ]Z = AX(∇Y (Z))−∇Y (AX Z) = (∇Y (AX))Z

Let’s fix a point o ∈ M and let Vo = To(M). Consider the space End(Vo)×Vo and endowit with a bracket operation

[(A,a),(B,b)] = ([A,B]−R(a,b),Ab−Ba+T (a,b))

This is not, in general, a Lie bracket, for the Jacobi Identity is not usually satisfied.However nothing prevents a subspace of End(Vo)×Vo to be a Lie algebra.

4 Let Y = ∑aiXi, Z = ∑b jX j be vector fields of M. We have:

T (Y,Z) = ∇Y Z −∇ZY − [Y,Z]

= ∑i, j

ai∇Xib jX j −b j∇X j aiXi − [aiXi,b jX j]

= ∑i, j

aib j(∇XiX j −∇X j Xi − [Xi,X j]

)thus, T (Y,Z)p depends only on Yp and Zp


Lemma 2.3.15. Suppose that M has a smooth linear connection ∇ such that ∇ is invariant byAutloc(σ). Consider g′ the space of germs of Killing vector fields at o ∈ M (it is of course, a Liealgebra) and consider the map

θo : g′ → End(Vo)×Vo

X ↦→((AX)o,Xo

)Then, θo is injective and preserves the bracket operation. In particular, its image is a Lie

algebra.

Proof. First notice that the hypothesis on the connection ∇ implies that the Killing fields areinfinitesimal affine transformations.

To prove that θo is injective we shall show that the germ of a Killing vector field X at o

depends only on (AX)o and Xo. To see this, consider a smooth curve γ(t) on M passing througho. We denote by a(t) = AX(γ(t)), v(t) = γ(t) and x(t) = X(γ(t)). From the Corollary 2.3.14, andAXY =−∇Y X −T (X ,Y ), it follows that

∇v(t)a(t) = R(x(t),v(t))

∇v(t)x(t) =−T (x(t),v(t))−a(t)v(t)

We have therefore a Cauchy problem and thus, the germ of X at o depends only on theinitial data: (AX)o and Xo.

Now, we must show that θo preserves the bracket operation. First, observe that

A[X ,Y ]+∇[X ,Y ] = L[X ,Y ] = [LX ,LY ] = [AX +∇X ,AY +∇Y ]

= [AX ,AY ]+ [AX ,∇Y ]+ [∇X ,AY ]+ [∇X ,∇Y ]

= [AX ,AY ]+ [AX ,∇Y ]+ [∇X ,AY ]+R(X ,Y )+∇[X ,Y ]

On the other hand [∇X ,AY ] = ∇X(AY ) and thus:

A[X ,Y ] = R(X ,Y )+ [AX ,AY ]+∇X(AY )−∇Y (AX)

Now suppose that X and Y are Killing vector fields. In particular, X and Y are infinitesimalaffine transformations and from Corollary 2.3.14, we have:

R(X ,Y ) = (∇Y (AX))

and thus

A[X ,Y ] = [AX ,AY ]+R(Y,X)


Now, as [X ,Y ] = ∇XY −∇Y X −T (X ,Y ) and

AXY −AY X = ∇XY +T (Y,X)−∇Y X −T (X ,Y ) = [X ,Y ]+T (Y,X)

we have

θo([X ,Y ]) =((A[X ,Y ])o, [X ,Y ]o

)=([AX ,AY ]o +R(Y,X)o,(AXY )o − (AY X)o −T (X ,Y )o

)=[((AX)o,Xo),((AY )o,Yo)

]= [θo(X),θo(Y )]

Which finishes the proof.

Remark 2.3.16. Notice that this lemma gives an upper bound on the dimension of the Liealgebra of Killing vector fields if there exists a compatible connection.

39

CHAPTER

3CONTACT ANOSOV ACTIONS

In this section we define Anosov actions of abelian groups and present a few alreadyknown results. Our goal is to study Anosov action associated with a specific geometric struc-ture, the generalized k-contact structure (Subsection 3.2.1). With this goal in mind, we presentsome related definitions1 already present in the literature: Contact Pairs (Subsection 3.2.2.1),r-Contact Structures (Subsection 3.2.2.2), Multicontact Structures (Subsection 3.2.2.3), Poly-contact Structures (Subsection 3.2.2.4) and Pluricontact structures (Subsection 3.2.2.4). Thefollowing diagram clarifies which condition implies which:

r−Contac Strucute ⇐ Contact Pair of even type

⇓

Contact Pair ⇒ Generalized k−Contact Structure ⇒ Multicontact Structure

⇓

Polycontact Structure ⇒ Pluricontact Structure

We also give sufficient conditions (Proposition 3.2.56) for a Pluricontact structure to alsobe a generalized k-contact structure.

3.1 Anosov actions

Definition 3.1.1. Consider a compact smooth manifold M and a smooth action φ : Rk ×M → M.This action is said to be Anosov if there exists an element a ∈ Rk, called an Anosov element,1 For the sake of clarity, we took the liberty of renaming some of these definitions.

40 Chapter 3. Contact Anosov actions

such that φa acts on M normally hyperbolically, that is, there exists a, dφ a invariant, continuous,splitting T M = E+⊕T φ ⊕E−, where T φ is the distribution tangent to the orbits, such that, thereexists positive constants C,λ for which

‖dφta(u+)‖ ≤Ce−tλ‖u+‖ ∀t > 0 ∀u+ ∈ E+ (3.1)

‖dφta(u−)‖ ≤Cetλ‖ui‖ ∀t < 0 ∀u− ∈ E− (3.2)

Remark 3.1.2. Notice that as M is compact, the estimates (3.1) do not depend on the choiceof metric (that is, the constants may depend on the metric but the subbundles still contract andexpand uniformly)((HIRSCH; PUGH; SHUB, 1970)).

Remark 3.1.3. The set A (φ) of Anosov elements forms an open set of Rk, and each connectedcomponent of A (φ), is an open cone. Moreover, Anosov elements on the same open cone willhave the same invariant distributions ((BARBOT; MAQUERA, 2013)).

Theorem 3.1.4 ((HIRSCH; PUGH; SHUB, 1970)). The bundles E+, E−, E+⊕T φ , and T φ ⊕E−

are integrable, with smooth leaves. The resulting foliations F ss, F uu, F ws, F wu, are calledstrong stable, strong unstable, weak stable and weak unstable foliations. In general, thosefoliations are not smooth, they are simply Hölder continuous.

Definition 3.1.5. A group action G yX is said to be topologically transitive if it admits a denseorbit. If Y ⊂ X , we say that the action is topologically transitive on Y if Y is G-invariant andthere exists an orbit which is dense in Y .

Theorem 3.1.6 (Spectral decomposition for Anosov actions (ARBIETO; MORALES, 2009)).Let A : Rk → Di f f (M) be an Anosov action. Then, the non-wandering2 set NW (A) of A is afinite union NW (A) = Ω1 ∪ . . .Ωl of disjoint compact and invariant sets Ωi. Moreover, each Ωi

cannot be further subdivided in two compact, non-empty, disjoint and invariant subsets and theaction A on each Ωi is topologically transitive on Ωi.

Corollary 3.1.7. An Anosov action φ : Rk ×M → M that preserves a volume is topologicallytransitive.

Proof. Let x ∈ M and let K ⊂ Rk be a compact set, suppose that K ⊂ BR(0) ⊂ Rk, and letg ∈ BR(0)c. Then, φ(g, ·) defines a volume preserving diffeomorphism of M. Fix an arbitraryneighbourhood U of x. From the Poincaré recurrence theorem, for almost every y∈U , there existsn∈N such that φ(n ·g,y)∈U . In particular, (n ·g) ·U∩U = /0. As (n ·g)∈BnR(0)c ⊂BR(0)c ⊂Kc

we conclude that x is non-wandering. Thus, NW (φ)=M. From the connectedness of M, it followsfrom Theorem 3.1.6, that φ is topologically transitive.2 Let G → Hom(M) be an action of a topological group G on a topological manifold M. We shall say

that a point x ∈ M is non-wandering, if, for every neighbourhood U of x and for every compact subsetS ⊂ G, there exists g ∈ Sc such that gU ∩U = /0. The set of non-wandering points is clearly invariantby the action of G.

3.2. Generalized k-contact Structure 41

3.2 Generalized k-contact Structure

In this subsection we present the definition and basic properties of a generalized k-contact structure, and what means a generalized k-contact Anosov action. For the sake ofcompleteness, we also present the definition and some relevant properties of two stronger(and previously known, (BANDE; HADJAR, 2005), (MONTANO, 2017),(BOLLE, 1996))structures: Contact pair structures (introduced and studied by Gianluca Bande and AmineHadjar (BANDE; HADJAR, 2005)) and k-contact structures (studied by Philippe Bolle (BOLLE,1996), Beniamino Cappelletti (MONTANO, 2017), Adriano Tomassini and Luigi Vezzoni(TOMASSINI; VEZZONI, 2008), among others).

Afterwards, we also explore two (weaker) possible ways to generalize the definition ofcontact structures (Multicontact structures (VITAGLIANO, 2015) and Polycontact structures(APOSTOLOV et al., 2017) and (ERP, 2010)), and how they relate to the generalized k-contactstructure.

We point out some differences between those definitions and briefly expose the contextin which they arose.

3.2.1 Generalized k-contact structures

Definition 3.2.1. A generalized k-contact structure on a manifold M of dimension 2n+ k is acollection of, k non zero, linearly independent, 1-forms α1, . . . ,αk and a splitting T M = I⊕F ,dimI = k, such that, for every 1 ≤ j ≤ k we have,

1. F =⋂k

i=1 kerαi

2. dα j|F is non degenerate;

3. ker(dα j) = I

We denote this structure by (M,α,T M = I ⊕F).

Remark 3.2.2. It follows from the definition that, for every j = 1, . . . ,k α1 ∧·· ·∧αk ∧dαnj is a

volume form.

Remark 3.2.3. As dim(F) = 2n, the condition

dα j|F is non degenerate,

is equivalent to

dαnj |F = 0


Lemma 3.2.4. The 1-forms that define a generalized k-contact structure have constant rank3

2n+1

Proof. As dimI = k we have dim(F) = 2n, and the second item implies that dα j(p)n = 0 foreach point p ∈ M. As ker(dα j) = I and T M = I ⊕F then, dα

n+1j (p) = 0 for every point p ∈ M.

Thus, rank(α j) on each point is either 2n or 2n+1 Finally, as (α j)|kerdα j = 0 then α j has oddrank everywhere, that is, rank(α j) = 2n+1 at every point.

Lemma 3.2.5. For each j, there is a unique vector field X j ∈ Γ(M, I) such that αi(X j) = δi j.These vector fields are called Reeb vector fields.

Moreover, the Reeb vector fields commute one with each other:

[Xi,X j] = 0

Proof. By definition, the characteristic space of each αi is contained in I . As (α1∧·· ·∧αk)|I =0, then, for each i = 1, . . . ,k,

I ∩⋂

ker j =i α j = 0

A dimension analysis shows that in fact dim(I ∩⋂

ker j =i α j) = 1 and thus there exists for eachpoint p ∈ M a unique vector X j(p) = 0 such that iX jdαi = 0, and iX jαi = δi j.

To see that they commute, we shall show that

i[Xi,X j]α1 ∧·· ·∧αk ∧dαnl = 0

As α1 ∧·· ·∧αk ∧dαnl is a volume form, this means that [Xi,X j] = 0.

First we notice that for any i, l we have dαi ∧dαnl = 0, and thus, for any 1 ≤ j ≤ k we

haved(α j ∧dα

nl ) = 0

and more generally, for any 1 ≤ j1 < .. . < js ≤ k we have

d(α j1 ∧·· ·∧α js ∧dαnl ) = 0

Thus, for any 1 ≤ j ≤ k

(d ∘ iX j)α1 ∧·· ·∧αk ∧dαnl = d(α1 ∧·· ·∧α j−1 ∧α j+1 ∧·· ·∧αk ∧dα

nl ) = 0

and similarly(d ∘ iX j ∘ iXi)α1 ∧·· ·∧αk ∧dα

nl = 0

3 Remember that the rank of a differential form ω of rank p at a point y is the co-dimension of thecharacteristic space

C (ω)(y) = X ∈ TpM ; iX ω(y) = 0 and iX dω(y) = 0


Moreover, as α1 ∧·· ·∧αk ∧dαnl is a volume form, we have

d(α1 ∧·· ·∧αk ∧dαnl ) = 0

Thus, using Cartan’s formula4:

i[Xi,X j]α1 ∧·· ·∧αk ∧dαnl = [iA,LB]α1 ∧·· ·∧αk ∧dα

nl

= [iA, iB ∘d +d ∘ iB]α1 ∧·· ·∧αk ∧dαnl = 0

Remark 3.2.6. Thus, a generalized k-contact structure comes with a canonical Rk action φ ,called a generalized k-contact action. It is also clear that I is precisely T φ , the distribution tangentto the action. Moreover, if the splitting T M = I ⊕F is smooth, then so are the vector fields X j

and therefore so is the action φ .

It is sometimes convenient to denote a generalized k-contact structure on M as the 4-tuple(M,α,φ ,F), where α = (αq, . . . ,αk), φ denotes the canonical generalized k-contact action andF ≤ T M is the φ -invariant subbundle where dα j is non degenerate.

Example 3.2.7. There exist an easy way to build examples of generalized k-contact manifoldsfrom an existing one. Let (B,α,T B = I⊕F) be a generalized r contact manifold and π : M → B

a principal torus bundle (with standard fiber Tl). Let Yi, i = 1, . . . , l be the vector fields thatgenerates the Tl-action. Consider on each torus, fibers of the bundle, the canonical 1-formsξ1, . . . ,ξl , ξi(Y j) = δi j. Consider a flat connection on M, and let T M = H ⊕V the associateddecomposition into horizontal (H) and vertical (V ) bundles. Using this connection we can extendthe forms ξ1, . . . ,ξl to global forms on M.

In fact, the as the bundle π : M → B is a principal torus bundle, the vertical part V of thesplitting T M = H ⊕V has a canonical splitting V = RY1 ⊕·· ·⊕RYl given by the Tl action onM. We define the forms ξ1, . . . ,ξl by

ξi(Yj) = δi j ; ξi(H = 0)

As the vertical distribution is involutive, it follows that dξi(Yl,Yk) = 0 for any l,k. It isknown (Nesterov, A. (NESTEROV, 2000) Lemma 3.12) that for any horizontal vector field Y ,

4 The Cartan’s formula is LX = iX ∘ d + d ∘ iX where LX denotes the Lie derivative. A classicalconsequence (Kobayashi, S and Nomizu, K. (KOBAYASHI; NOMIZU, 1963), Section I.3 Proposition3.10) of Cartan’s formula is the identity

i[A,B] = [iA,LB]


the commutator [Yi,Y ] is horizontal. Therefore, the 1-forms ξi satisfies, for any horizontal vectorfield W ,

dξi(Yl,W ) = Yl(ξi(W )︸︷︷︸=0

)−W (ξi(Yl)︸︷︷︸=1

)−ξi([Yl,W ])︸︷︷︸=0

= 0 (3.3)

and thus iYl dξ = 0. Moreover, as the connection is flat, the horizontal distribution is involutiveand therefore, for any two horizontal vector fields W1,W2 we have

dξi(W1,W2) =W1(ξi(W2))−W2(ξi(W1)−ξi([W1,W2]) = 0 (3.4)

As we have the splitting T B = I ⊕F , the fibres Hp of the horizontal bundle H can beidentified with the fibres Tπ(p)B of the tangent bundle T B, we have an induced splitting H = I⊕ F .We denote V =V ⊕ I ⊂ T M. From equations 3.4 and 3.3 we obtain that I ⊂ kerdξi and thus, itis clear that, for any ( j1, . . . , jl) ∈ 1, . . . ,rl , we have the following generalized (l + r)-contactstructure on M:

(M,ξ1 +π*α j1, . . . ,ξl +π

*α jl ,π

*α1, . . . ,π

*αr,T M = V ⊕ F)

In particular, if B is a contact manifold (equivalently, 1-contact or generalized 1-contactmanifold), and M → B is a principal torus bundle with a flat connection, then, the inducedstructure we have constructed above is a (l +1)-contact manifold.

Lemma 3.2.8. Let (M,α,T M = I⊕F) be a generalized k-contact structure, and let B = (βi j) ∈Mk×k(R). We define ηi = αi −∑

kj=1 βi jα j. Then, if B is sufficiently small, (M,η ,T M = I ⊕F)

is a generalized k-contact structure.

Proof. As B is small, then Id −B is invertible, then⋂ker(ηi) =

⋂kerαi = F

Also, we can writedηi = (1−βii)dαi +∑

j =iβi jdα j

where βi j are small. And thusdη

ni = dα

ni +ω

where ω is small. As non degeneracy is an open condition, if ω is small enough, then dηi is nondegenerate. Similarly, dηn

i = dαni +ω0 where ω0 is small, and therefore

η1 ∧·· ·∧ηk ∧dηnj = det(Id −B)α1 ∧·· ·∧αk(dα

ni +ω0)

which is non zero for small ω0.

Finally, I ⊂ ker(dη j), but as T M = I ⊕F and dη j is non degenerate on F (for smallperturbation of the identity B), then ker(dη j)⊂ I


A similar argument proves the following Lemma

Lemma 3.2.9. Consider a manifold M of dimension 2n+ k, a splitting, T M = I ⊕F , dimI = k

and, linear independent, non vanishing 1-forms α1, . . . ,αk such that

∙ F =⋂

i ker(αi)

∙ dαk is non degenerate on F and I = ker(dαk)

∙ I ⊂ kerdα j for j = 1, . . . ,k−1

Then, there exists B = (βi j) ∈ GL(Rn−1) and a change of coordinates η j = ∑i βi jα j , j =

1, . . . ,k−1 such that (M,η1, . . . ,ηk−1,αk,T M = I ⊕F) is a generalized k-contact manifold

Remark 3.2.10. Notice that under the conditions of the Lemma 3.2.9, we have well definedvector fields X1, . . . ,Xk tangent to I such that α j(Xi) = 1. The same arguments of Lemma 3.2.5shows that these vector fields commute with each other, and therefore defines an action of Rk onM.

With some additional conditions, we can improve Lemma 3.2.8

Lemma 3.2.11. Under the conditions of the previous Lemma 3.2.8, suppose that the action istopologically transitive. For B ∈ Gl(Rk), denote by η = Bα the change of coordinates ηi =

∑ j βi jα j. Then, there exists a Zariski open subset of Λ ⊂ Gl(Rk) such that (M,Bα,T M = I⊕F)

is generalized k-contact.

Proof. Like in the previous lemma, we have

E =k⋂

i=1

αi =k⋂

i=1

ηi

and

I ⊂ kerdη j ∀ j = 1, . . . ,k

we must show that for a Zariski open set Λ, the conditions

∙ dη j|F is non degenerate for every 1 ≤ j ≤ k;

∙ η1 ∧·· ·∧ηk ∧dηnj is a volume form.

are satisfied.

Now, denote

dαJ = dα

J11 ∧·· ·∧dα

Jnk


for any multi-index J = (J1, . . . ,Jk) ∈ Nk. Then, we can write

dηnj = ∑

|J|=nQJdα

J

where QJ is some polynomial in the variables βi j and |J|= J1 + · · ·+ Jk

Notice that dαJ is clearly a top form over F , and thus dαJ = fJdαnk for some function

fJ . As the action is topologically transitive, and the forms dαJ and dα1 are invariant by thisaction, this function is constant over M. Thus, we write

dηnj = Pj(B)dα

nk

where Pj(B) is a polynomial on the coefficients βi j of B. We also write

η1 ∧·· ·∧ηk ∧dηkj = Pj(B)det(B)α1 ∧·· ·∧αk ∧dα

nk

And thus, conditions 3.2.1 are satisfied when Pj(B) = 0. This polynomial is non zero, forPj(Id) = 1, and thus, the conditions 3.2.1 are met for B in a Zariski open set.

Remark 3.2.12. We can understand the re-parametrization η = Bα as a re-parametrization ofthe action, that is, let (M,α,T M = I ⊕F) be a generalized k-contact structure, X1, . . . ,Xk theassociated vector fields. For every j, we denote by φ j the flow associated with the vector fieldX j. Let e1, . . . ,ek be the standard basis on Rk. The associated action φRk yM is given by

φ(e j) = φj

Let B = (βi j) ∈ GL(Rk) and f j = ∑ki=1 βi jei be another basis on Rk and Yj = φ( f j). Let

us define 1-forms on M byη j(Yi) = δi j ; η j(E) = 0

thenη = B−1

α

Definition 3.2.13. A generalized k-contact action φ will be called generalized k−contact Anosov,if the action is Anosov.

Lemma 3.2.14. If φ : Rk → M is a generalized k-contact action, then there exists B ∈ GL(Rk)

such that every Reeb vector field is Anosov and they have the same invariant splitting T M =

T φ ⊕E+⊕E−

Proof. As the action preserves the volume form α1 ∧·· ·∧αk ∧dαnj , the action is topologically

transitive (Corolary 3.1.7). Thus, the previous lemma (Lemma 3.2.11) implies that for almost anylinear re-parametrization of the action, the corresponding structure is still generalized k-contact.It is known (Remark 3.1.3) that the set of Anosov elements is a disjoint union of connected opencones, and moreover, the invariant splitting T M = T φ ⊕E+⊕E− depends only on the choice ofopen cone, thus just choose a re-parametrization that puts every Reeb vector field in the sameopen cone.


Definition 3.2.15. Let (M,α,T M = I ⊕F,φ) be a generalized k-contact Anosov action. Theparametrization α will be called adapted if every Reeb vector field is Anosov.

Remark 3.2.16. The Lemma 3.2.14 proves that every generalized k-contact Anosov actionadmits an adapted parametrization.

Remark 3.2.17. Another way to see this definition is to start with an Anosov Rk-action φ ,consider the action as given by a family of commuting vector fields X j, where each X j is thevector field associated with an Anosov element. Using the splitting T M = T φ ⊕E+⊕E− wedefine the dual 1-forms α1, . . . ,αk, and we suppose that each of those forms have constant rank2n+1 and satisfies:

α1 ∧·· ·∧αk ∧dαnj is a volume form.

Remark 3.2.18. For the rest of this text, α1, . . . ,αk will always denote the 1-forms associatedwith an adapted parametrization of a generalized k-contact Anosov action, and X1, . . . ,Xk theassociated dual Anosov vector fields. We shall also denote each volume form dM j = α1 ∧·· ·∧αk ∧dαn

j .

Remark 3.2.19. Notice that dimE+ = dimE− = n. This follows from the fact that, for somefixed j, dα j restricted to E+⊕E− is a φ - invariant symplectic form, the hyperbolic dynamicswill ensure that E± are Lagrangian subspaces.

Lemma 3.2.20. The distribution F in the generalized k-contact structure is non integrable.

Proof. Suppose that F is integrable. From Frobenius Theorem, this is equivalent to [F,F ]⊂ F .Take two vector fields Z,W ∈ Γ(M,F), in particular, we have

α j(Z) = α j(W ) = α j([Z,W ]) = 0

and thus

dα j(Z,W ) = Z(α j(W ))−W (α j(Z))−α j([Z,W ]) = 0

which contradicts the fact that dα j is non degenerate on F .

Remark 3.2.21. The distribution F is maximally non integrable in the following sense: Forevery vector field Z tangent o F there exists a vector field W also tangent to F such that [Z,W ] isnot tangent to F .

Remark 3.2.22. Consider, on a manifold M of dimension 2n + k, a distribution F ⊂ T M,dimF = 2n, and let J be the C∞(M,R)-submodule of 1-forms on M that vanishes on F .

Let J k denote the space of k-forms of the type ξ1 ∧·· ·∧ξk with ξ j ∈ J , (dJ )(n) thespace of 2n-forms of the type dη1 ∧·· ·∧dηn, η j ∈ J, (dJ )n the space of 2n-forms of the type(dη)n, η ∈ J.


Let J k ∧ (dJ )n (resp. J k ∧ (dJ )(n)) denote the space of 2n+ k-forms of the typeω ∧Ω, where ω ∈ J k and Ω ∈ (dJ )n (resp. Ω ∈ (dJ )(n))

We can compare the notions of k-contact structure and generalized k-contact structure inthe following way:

∙ If F is the 2n-plane distribution of a k-contact structure, then, every non-zero element ofJk ∧ (dJ)(n) is a volume form.

∙ If F is the 2n-plane distribution of a generalized k-contact structure, then, there existsnon-zero element of Jk ∧ (dJ)n which is a volume form.

The first item follows from the fact that J = SpanC∞(M,R)α1, . . . ,αk, therefore Jk =

f α1 ∧·· ·∧αk ; f ∈C∞(M,R∖0) and Jk+1 = 0.

Moreover dJ = f dα1 + d f ∧η ; f ∈ C∞(M,R),η ∈ J, and thus, if dη ∈ dJ, thenα1 ∧·· ·∧αk ∧dη = f α1 ∧·· ·∧αk ∧dα1.

By induction we see that α1 ∧·· ·∧αk ∧dη1 ∧ . . .dηn = f1 · · · · · fnα1 ∧·· ·∧αk ∧dαn1 .

The second item is clear.

3.2.2 Other contact like structures

3.2.2.1 Contact Pairs

The notion of contact pair was introduced by Bande, G. And Hadjar, A. to study certainprincipal torus bundles on surfaces.

Definition 3.2.23. A contact pair (C.P.) on a smooth manifold M of dimension 2h+2k+2 is acouple of 1-forms (α,η) satisfying:

∙ dαh+1 = dηk+1 = 0 for some positive integers h,k.

∙ α ∧dαh and η ∧dηk vanish nowhere.

∙ Ωα,η = α ∧dαh ∧η ∧dηk is a volume form

This contact pair is said to be of type (h,k). We shall usually denote Ω = Ωα,η when there is noconfusion.

Example 3.2.24. The easiest way to construct a manifold with a contact pair is to multiplyingtwo contact manifolds: If (M2h+1

1 ,α) and (M2k+12 ,η) are contact manifolds (α and η their


respective contact forms), then we can consider α and η as 1-forms on the product M1 ×M25. It

is clear that (α,η) is a contact pair on M1 ×M2 of type (h,k).

Another construction given by Bande. G and Hadjar, A.(BANDE; HADJAR, 2005), isby taking a contact manifold (M2h+1,α) with a contactomorphism f . The suspension manifoldM × [0,1]/(x,1) ( f (x),0) is a contact manifold of type (h,0). Bande. G and Hadjar, A. alsoconstruct examples on nilpotent Lie groups of dimension 4 and 6.

Remark 3.2.25. Notice that, if M is a manifold with a Contact Pair (α,η) of type (h,k) then

(M,α +η ,α −η ,T M = (ker(dα)∩ker(dη))⊕ (kerα ∩kerη)

is a generalized 2-contact manifold. In this sense, generalized k-contact structures generalize thenotion of Contact Pairs, in particular, we have the following lemma.

Lemma 3.2.26 (Bande, G. Hadjar, A. (BANDE; HADJAR, 2005), Theorem 2.2). We canassociate to a contact pair (α,η) on M vector fields, called Reeb vector fields, Xα and Xη

characterized by:

∙ iXαΩα = iXη

Ωη = 0

∙ α(Xα) = η(Xη) = 1

Those vector fields are unique. Moreover, [Xα ,Xη ] = 0, and both α and η are Xα and Xη

invariant.

Which means that associated with a contact pair, there is an action of R2, and the contactpair is invariant by this action.

3.2.2.2 r-Contact structures

The notion of r-contact manifolds was used by Philippe Bolle in (BOLLE, 1996) to studycertain class of co-isotropic submanifolds6 of a symplectic manifold. The r-contact conditionwas used to generalize the well known contact condition7,8 . Like in the usual contact case, we5 We consider the projections π j : M1 ×M2 → M j and consider α = π*

1 (α) and η = π*2 (η). We abuse

notation and set α = α and η = η6 Given a symplectic manifold (M,ω), a submanifold S of co-dimension p is called co-isotropic if

dim(ker ω|S) = p everywhere on S7 For a given symplectic manifold (M,ω), a co-dimension 1 submanifold S is of contact type if S is a

contact manifold (S,α), co-isotropic and dα = ω|S8 Philippe Bolle’s motivation is that such co-isotropic manifolds admits a canonical Rk-action, and his

goal was to prove that under certain conditions, there is a non trivial element X of Rk, whose associatedflow admits a non trivial, periodic, non contractible orbit. This is result towards a generalization ofWeinstein Conjecture, which propose the existence of a such orbits for a Reeb flow on an arbitrary,compact, contact manifold.


can forget the ambient manifold and consider the k-contact condition as a geometric structure onits own. In this case, the r-contact is an extension of the usual contact structure in the sense thatfor r = 1 we obtain the traditional contact structure.

Definition 3.2.27. A manifold M of dimension 2n+ r is said to be r-contact, if there existsnowhere zero, linearly independent, 1-forms α1, . . . ,αr such that

1. dαi = dα j for every 1 ≤ i, j ≤ r.

2. dαn+1i = 0

3. α1 ∧·· ·∧αk ∧dαni is a volume form on M.

We denote it by (M,α1, . . . ,αr), or simply (M,α).

Remark 3.2.28. Philippe Bolle was in a situation with an ambient symplectic manifold, andhe studied co-isotropic submanifolds. The first condition is rewritten as dαi = ω|M where ω

is the symplectic form on the ambient manifold. Together with the fact that he is interested inco-isotropic submanifolds, this modification means that the second condition was superfluous(and in fact it was not included in Bolle’s definition).9

Remark 3.2.29. It is easy to see that generalized k-contact structures are in fact generalizationsof r-contact structures. Moreover, if M is a manifold with a Contact Pair (α,η) of type (h,k)

and, either k or h is even, then

(M,α +η ,α −η)

is a 2-contact manifold.

Lemma 3.2.30 (Philippe Bolle, (BOLLE, 1996)). For a given r-contact manifold M, there existunique, linearly independent, vector fields X1, . . . ,Xr such that

∙ α j(Xi) = δi j, for every i, j.

∙ iXidα j = 0 for every i, j.

These vector fields are called Reeb vector fields and they satisfy:

[Xi,X j] = 0 ∀i, j

9 To be precise, Bolle considers a symplectic manifold (N,ω) of dimension 2(n+ r), and is interestedin co-isotropic study submanifolds M of co-dimension r, that is, dim(kerω|M) = r everywhere. Hedefined a submanifold M to be of r-contact type if it is co-isotropic, and has r, linearly independent1−forms α1, . . . ,αr such that dαi = ω|M and α1 ∧ ·· · ∧αk ∧ωn

|M is a volume form. The co-isotropy

condition implies that ωn+1|M ≡ 0.


Remark 3.2.31. Other authors who studied the r-contact manifolds ((TOMASSINI; VEZZONI,2008),(MONTANO, 2017)) do not ask explicitly for the second condition (dα

n+1i = 0), but it is

a necessary condition to obtain the Reeb vector fields above10 not explicitly stated by the authors.For example, consider R5 with coordinates (x1,y1,x2,y2,z) and consider

Σ = y1dx1 + y2dx2

α1 = dz+dx1 +Σ

α1 = dy1 +dx1 +Σ

α1 = dy1 +Σ

It is clear that dα j = dΣ for every j, and a quick computation shows that

α1 ∧α2 ∧α3 ∧dΣ = dz∧dx1 ∧dy1 ∧dx2 ∧dy2

However ker(dΣ) = Span∂z. This means that it is impossible to have 3 linearly independentvector fields X1,X2,X3 such that αi(X j) = δi j and Xi ∈ ker(dα j).

3.2.2.3 Multicontact

Luca Vitagliano introduced the notion of multicontact structure as the higher co-dimensioncounterpart of multisymplectic11 structure. It was previously known (Rogers (ROGERS, 2012))that there is an L∞-algebra12 P which is the multisymplectic structures analogue of the Poissonalgebra13. Vitagliano constructed, for the multicontact structure an L∞-algebra J which is thehigher dimensional analogue of the Jacobi bundle14, that is, this J and M interacts in a similar

10 For example, Cappelleti (MONTANO, 2017) on page 2. clearly states the existence of r linearlyindependents vector fields ξ1, . . . ,ξr such that α j(ξi) = δi j and ξi ∈ kerdα j

11 A closed (n+1)-form ω on M is called pre-n-plectic if ker(ω) has constant rank. If ker(ω) = 0everywhere, ω is called a n-plectic form, and defines a multiplectic structure on M. (Cantrijn et al.(CANTRIJN; IBROT; LéON, 1999))

12 Recall that a L∞ algebra is a Z-graded vector space V and multi-linear maps fn : V⊗n →V of degreen−2 called brackets that satisfies a (graded) skew symmetry property and a (strong homotopy) Jacobiproperty. For more details see (STASHEFF, )

13 A Poisson algebra is a Lie algebra (V, [ · , · ]) where for each a ∈V , [a, · ] : V →V is a derivation of V .The algebra of smooth function on a symplectic manifold, endowed with the Poisson bracket is anexample of a Poisson algebra

14 A Jacobi bundle on M is a line bundle E → M equipped with a Lie bracket on its local sections. TheReeb vector field X associated with a contact manifold (M,α) gives a trivial line bundle and for anytwo sections f X and gX of this bundle, the map bellow defines a Lie bracket on the space of sections

f ,g= dg(Λ(d f ))+ f dg(X)−gd f (X)

where Λ : T *M → T M is defined as

Λ(ξ )α = 0 and Λ(ξ )(dα) = ξ (X)α −ξ


way as the Jacobi bundle and Poisson algebra.

Consider a manifold M and E ⊂ T M a linear subbundle. Let us denote by N the quotientbundle T M/E. We have, for any open set U ⊂ M the following exact sequence:

0 → Γ(U,C)→ Γ(U,T M)→ Γ(U,N)θ→ 0

Where θ denotes the natural projection map. We also have a well defined bilinear map called thecurvature form of E,

B : Γ(M,E)×Γ(M,E)→ Γ(M,N)

(Z,W ) ↦→ θ([Z,W ])

Lemma 3.2.32. The curvature form B is tensorial, that is, it induces a bundle map

B : Λ2E → N = T M/E

Proof. Let us show that for any p ∈ M and local vector fields Z,W , in neighbourhood U of p,tangents to E, the value B(Z,W )(p) depends only on the values Z(p) and W (p).

In fact, consider Z′,W ′ vector fields on U such that Z′(p) = Z(p) and W ′(p) = W (p).Then, we write Z′ = f Z and W ′ = gW for (smooth) functions f ,g such that f (p) = g(p) = 1. AsZ and W are tangent to E, it follows that θ(Z) = θ(W ) = 0 and, thus, we have

B(Z′,W ′) = θ([ f Z,gW ])

= θ(Z(g)W −W ( f )Z + f g[Z,W ]

)= Z(g)θ(W )−W ( f )θ(Z)+ f gθ([Z,W ])

= f gθ([Z,W ])

in particular,

B(Z′,W ′)(p) = f (p)g(p)︸︷︷︸=1

θ([Z,W ])(p) = θ([Z,W ])(p) = B(Z,W )(p)

Lemma 3.2.33. Let M → Q be a G-principal bundle and E ⊂ T M is the horizontal space of aconnection on M. Then −B coincides with the usual g-valued curvature form, where g denotesthe Lie algebra of G, restricted to horizontal vector fields.

Proof. Let A ∈ Γ(M,T *M⊗g) denote a connection form on the G-principal bundle π : M → Q.The horizontal distribution E associated with this connection is defined (pointwise) by Ep =

ker(Ap) where Ap is the induced map

Ap = A(p) : TpM → g


This allow us to identify TpM/Ep = g.

The usual curvature form is defined as ΩA = dA+A∧A. Thus, for horizontal vectorfields Z,W we have

ΩA(Z,W ) = dA(Z,W )+ [A(Z),A(W )]

= Z(A(W ))−W (A(Z))−A([Z,W ]) =−A([Z,W ])

and thus

ΩA(Z,W )(p) =−A([Z,W ])(p) =−[Z,W ](p) ModEp =−θ([Z,W ])(p) =−B(Z,W )(p)

as we desired.

The characteristic distribution associated with E is the distribution

C (E) = Y ∈ Γ(M,E) ; B(Y, · ) = 0

Definition 3.2.34 (Multicontact structure: Luca Vitagliano (VITAGLIANO, 2015)). A distri-bution E is called pre-multicontact if the characteristic distribution C (E) has constant rank. IfC (E) = 0, E is called a multicontact distribution.

Remark 3.2.35. Notice that the characteristic distribution can be defined locally, that is, we canconsider, for any open set U ⊂ M, the local distribution CU(E) of local vector fields Y ∈ Γ(U,E)

such that BU(Y,Z) = 0 for any Z ∈ Γ(U,E). Then, a vector field Y ∈ C (E) if, and only if, forany open set U , the restriction of YU of Y to U belongs to CU(E).

Lemma 3.2.36. If (M,α,T M = I ⊕F) is a generalized k-contact structure, then (M,F) is amulticontact structure.

Proof. As dα j is non degenerate on F , for any non vanishing vector field Z tangent to F , thereexists another non vanishing vector field W tangent to F , such that dα j(Z,W ) is nowhere zero.

On the other hand,

dα j(Z,W ) = Z(α j(W )︸︷︷︸=0

)−W (α j(Z)︸︷︷︸=0

)−α j([Z,W ]) =−α j([Z,W ])

Moreover, the splitting T M = I ⊕F allow us to identify T M/F = I, and thus,

B(Z,W ) = ∑j

α j([Z,W ])X j

Therefore a vector field Z belongs to the characteristic distribution C (F) if, and only if,α j([Z,W ]) = 0 for every vector field W tangent to E and every j ∈ 1, . . . ,k.

As we already noticed, there is no such vector field and thus, the characteristic distributionis the zero section of T M.


3.2.2.4 Polycontact and Pluricontact

Van Erp introduced the notion of Polycontact to study a certain algebra of pseudo-differential operators. More precisely, he used the curvature form to associate with each tangentspace the structure of a 2-step nilpotent Lie algebra and understood that polycontact manifoldsare (in some sense) micro locally modelled after groups of Heisenberg type. Van Erp pointsout that the usual Heisenberg (pseudo differential) calculus on contact manifolds (developedby Taylor (TAYLOR, 1984)) is generalized to the polycontact context, defining the algebraΨ*

H(M) =⊕

d ΨdH(M) of Heisenberg pseudo differential operators15. Van Erp showed that the

existence of a projection with infinite dimensional kernel on Ψ*H(M) is equivalent to the non

degeneracy condition on the Levi form.

Definition 3.2.37 (Polycontact structure: Erik Van Erp (ERP, 2010)). A distribution E definesa polycontact structure, if for every open set U , and for any 1-form α ∈ Γ(U,N*), that doesn’tvanishes on U , the following bilinear map is non degenerate

Γ(U,E)×Γ(U,E)α∘BU→ R

where BU denotes the restriction of B to sections defined on U , and non degeneracy meansthat for any vector field Z ∈ Γ(U,E) and any point p ∈U , there exists W ∈ Γ(U,E) such thatαp(BU(Z,W )(p)) = 0.

Apostolov et al ((APOSTOLOV et al., 2017)) relaxed the contact condition further still,defining what we will call a pluricontact structure (they did not use this terminology, calling(M,E) a contact structure of co-dimension l and dimension n = rank(E)/2. We will use theterminology Pluricontact in accordance to Choi, W. and Ponge, R. (CHOI; PONGE, 2017)16).Their goal is to extend the theory of toric contact manifolds to the higher co-dimension. Afterdefining the appropriate higher co-dimension analogue of the moment polytope of a compacttoric symplectic manifold, which they called the Grassmanian momentum map, they give aclassification of the contact toric manifolds in terms of the cohomology of a sheaf on the imageof the Grassmanian momentum map.

Definition 3.2.38 (Pluricontact structure: Apostolov et al (APOSTOLOV et al., 2017)). Adistribution E defines a pluricontact structure, if for every p ∈ M, there exists α ∈ (T M/E)*

15 We notice here that on the usual contact case, the Darboux theorem says that there exists no localinvariants for contact manifolds, and thus, local (and microlocal) analysis on contact manifolds areequivalent to local (and microlocal) analysis on Heisenberg groups, which carries a natural contactstructure. In the case of polycontact, there exists no unique local model, but the Heisenberg type Liealgebra structure on each tangent space of the manifold M allows to extend the Heisenberg calculus tothe polycontact context.

16 On their work Choi, W. and Ponge, R study Carnot manifolds, that is, manifolds whose tangent spaceshave a graded Lie algebra structure. Polycontact and pluricontact structures naturally induces suchstructures, and were given as examples on (CHOI; PONGE, 2017).


such thatαp ∘Bp : Ep ×Ep → R

is non degenerate. Where Bp denotes the restriction of the curvature form B : Λ2E → T M/E toEp ×Ep.

Lemma 3.2.39 (Van Erp, (ERP, 2010), Lemma 3). The following are equivalent

∙ The distribution E defines a polycontact structure

∙ For every (local) 1-form α which vanishes on E, the restriction of dα to E is non degener-ate.

This lemma allow us to prove

Proposition 3.2.40. If a distribution E defines a multicontact structure, then it defines a poly-contact structure

Proof. In fact, let α ∈ Γ(U,N*) be a local 1-form that vanishes on E, and let Z be a vectorfield on U tangent to E which is non vanishing. As dα is non degenerate on E, it follows thatthere exists W , another vector field on U tangent to E and non-vanishing, such that dα(Z,W ) =

−α([Z,W ]) = 0. And thus, θ([Z,W ]) = 0, that is, CU(E) = 0 for every U , and thus C (E) =

0.

Corollary 3.2.41. If (M,α,T M = I ⊕F) is a generalized k-contact structure, then (M,F) is apolycontact structure and a pluricontact structure.

Proposition 3.2.42. Let (M,E) be a polycontact structure and suppose that,

[Condition A:] There exists a unidimensional distribution I ⊂ T M such that the image of BU

Lies on Γ(U, I)

. Let X be a non zero vector fields which generates this distribution.

Then, E is associated with a l-contact structure (in the sense of (BOLLE, 1996)).

Proof. Notice that the curvature form B was normalized so, for every section α of E0, therestriction of dα to Λ2E is precisely α ∘B.

As the image of B is tangent to I, we can write, for any vector fields Z,W tangents to E,B(Z,W ) = fZ,W X .

Let α1, . . . ,αl ∈ Γ(M,E0) such that α j(X) = 1 for every j and α1(p), . . . ,αl(p) arelinearly independent for every p ∈ M, then, for any vector fields Z,W tangents to E, and any i

we havedαi(Z,W ) = α j ∘B(Z,W ) = α j( fZ,W X) = fZ,W


and thus, dαi = dα j for any i, j ∈ 1, . . . ,s.

As dα1 = · · · = dαl are non degenerate on E, it follows that α1 ∧ ·· · ∧αl ∧ dα jn is avolume form on M as desired.

Remark 3.2.43. If we use the fact that the curvature form B is tensorial, we can rewrite theCondition A as

[Condition A’:] The map B : Λ2E → T M/E has constant rank 1.

In what follows (M,E) will always be a pluricontact structure. Our goal is to follow theexposition of Apostolov et al. (APOSTOLOV et al., 2017) and prove a sufficient and necessarycondition for the reciprocal of the Corolary 3.2.41, that is, when does a pluricontact structurealso defines a generalized k-contact structure.

Definition 3.2.44. Let and let E0 denote the annihilator of E, that is,

E0 = α ∈ T *M ; α(E) = 0

and let, for every open set U ⊂ M

UE(U) = α ∈ Γ(U,E0) α ∘B is non degenerate

The setUE =

⋃U

open⊂ M

αp ∈ E0 ; ∀α ∈ UE(U) ∀p ∈U

is called the non-degeneracy locus of E and is an open submanifold of the total space of thebundle E0 → M.

A section α : M → UE is called a contact form of E.

Remark 3.2.45. Using the notations and definitions of this section and Lemma 3.2.9, we canrewrite the definition of generalized k-contact structure as follows:

A generalized k-contact structure on M is a pluricontact structure (M,E) such that I = T M/

E is a trivial bundle and there exists α ∈ E0 = I* a contact form such that I = ker(dα).

Definition 3.2.46. A (local) vector field X is called an infinitesimal (local) pluricontactomor-phism for (M,E) if LX Γ(U,E)⊂ Γ(U,E). We denote the Lie algebra of infinitesimal pluricon-tactomorphisms by con(M,E)

Lemma 3.2.47 (Apostolov et al. (APOSTOLOV et al., 2017), Proposition 2). For any contactform α : M → UE , we have T M = E ⊕K α , where K α is the ortogonal complement of E viadα .

Definition 3.2.48. The distribution K α defined above is called the Reeb distribution of α .


Definition 3.2.49. A pluricontact action of a Lie algebra g on (M,E) is a Lie algebra homomor-phism K : g→ con(M,E). We define κ : g×M → T M by κ(g, p) = K(g)(p), and we set K g

the distribution given byK g

p = SpanRK(g)(p) ; g ∈ g

Definition 3.2.50. A pluricontact action K : g→ con(M,E) is said to be transversal if

∙ rankK g = l everywhere on M

∙ K g∩E is the zero section of T M.

Or, equivalently, if T M = E ⊕K g.

Lemma 3.2.51 (Apostolov et al. (APOSTOLOV et al., 2017)). For a transverse pluricontactaction K : g→ con(M,E), there exists a unique g-valued 1-form η = ηg on M satisfying

∙ ker(ηg) = E

∙ ηg(K(g)) = g ; ∀g ∈ g

This form is called the canonical g-form associated with K

Lemma 3.2.52 (Apostolov et al. (APOSTOLOV et al., 2017), Lemma 1). Let K : g→ con(M,E)

be a transverse pluricontact action and η = ηg be the canonical g-form associated with K. Let B

be the extension of the levi form B by zero to Γ(M,T M) = Γ(M,E)⊕Γ(M,K g).

Then, for any g ∈ g we have

(LK(g)η)(X)+ [g,η(X)]g = 0∀X ∈ Γ(M,T M)

dη = η ∘ B− 12

η ∧η

Definition 3.2.53. Let K : g→ con(M,E) be a transverse pluricontact action, and let η be thecanonical g-form associated with K. Let λ ∈ g* and define ηλ : M/E → E0 by

ηλ (p)(W ) := λ (ηp(W ))

so ηλ (K(g)) = λ (g), and thus

dηλ

|E = λ ∘dη|E = ηλ ∘B

For λ ∈ [g,g]0 ⊂ g*, we call the pair (E,dηλ

|E ) the induced Levi structure. It is said to benon degenerate if ηλ is a contact form.

Remark 3.2.54 (Apostolov et al. (APOSTOLOV et al., 2017), page 7). If α is a contact form of(M,E) such that α(K(v)) is constant for every v, then the Reeb distribution K α coincides withK g. In particular,

K ηλ

= K g


Definition 3.2.55. Let K : g→ con(M,E) be an injective, transverse, pluricontact abelian (Thatis, g=Rk) action. Let λ ∈ g*∖0. Then (g,λ ) is called a Levi pair if (E,dηλ

|E ) is non degenerate.

Proposition 3.2.56. Suppose that (M,E,K : Rk → con) is an abelian, injective, transverse,pluricontact action that admits a Levi pair (g,λ ). Then, it defines a generalized k-contactstructure on (M,E).

Proof. We have λ : Rk → R non zero. We choose a basis e1, . . . ,ek of Rk, such that λ (e1) = 1and λ (e2) = · · ·= λ (ek) = 0.

Now, consider the canonical Lie algebra valued 1-form η can be written as η =

(η1, . . . ,ηk), where each η j is an usual R-valued 1-form. For our choice of basis e1, . . . ,ek

of Rk, we denote X j = K(e j) ∈ con(M,E). The condition

ηg(K(g)) = g ; ∀g ∈ g

means

η j(Xi) = δi j

As K is injective, K g = SpanX1, . . . ,Xk is k-dimensional.

In particular, this means that η1, . . . ,ηk are linearly independent.

The condition ker(ηg) = E means

E =⋂

j

ker(η j)

The condition

(LK(g)η)(X)+ [g,η(X)]g = 0∀X ∈ Γ(M,T M)

means

LX jηi = dηi(X j, · ) = 0

We also have, for vector fields Z,W tangents to E, and p ∈ M

dηλ

|E (Z,W )(p) = ηλ ∘B(Z,W )(p)

= λ (ηp(B(Z,W )(p)))

= λ (η1p(B(Z,W )(p)), . . . ,ηkp(B(Z,W )(p))) = η1p(B(Z,W )(p)).

As dηλ

|E is non degenerate, this means that η1∘B is also non degenerate. But dη j = η j ∘B

and thus dη1 is non degenerate on E.

3.3. Examples and Counter-examples 59

The condition K g = SpanX1, . . . ,Xk means that K g ⊂ ker(dη j) for every j.

Finally, the decomposition T M = K g⊕E proves that

K g = ker(dη1)

Thus, from Lemma 3.2.9, there exists a change of coordinates α j = ∑kj=1 βi jηi such that

(M,α1, . . . ,αk,T M = K g⊕E) is indeed a generalized k-contact structure on M.

3.3 Examples and Counter-examples

3.3.1 An example: SO(k,k+n)/SO(n)

Our goal in this section is to give an explicit construction of a generalized k-contactAnosov action on SO(k,k+n)/SO(n) which is SO(k,k+n)-invariant, and therefore descendsto a generalized k-contact Anosov action on Γ∖SO(k,k + n)/SO(n) for any uniform latticeΓ ⊂ SO(k,k+n), acting freely on SO(k,k+n)/SO(n). Latter on, we give a more geometric viewof this objects.

The algebraic nature of our example will allow us to generalize this construction onTheorem 3.3.26.

This examples are actually the first examples of generalized k-contact Anosov actions wefound before realizing that the general Theorem 3.3.26 is true. Moreover, using the machineryand notations used in the first part, we shall also prove (on sections 3.3.1.3 and 3.3.1.4) thatthe associated action does not come from a Contact Pair (in the case of k = 2) or a k-contactstructure (in the case of general k ≥ 2).

Before we construct the examples, we define the notion of algebraic action and give acriterion for an algebraic action be Anosov.

Definition 3.3.1. An algebraic (nilpotent) action is given by a quadruple (G,K,Γ,h), where

∙ G is a connected Lie group with Lie algebra g

∙ K is a compact subgroup of G with Lie algebra k

∙ h is a nilpotent subalgebra of g, contained in the normalizer Ngk of k and such thatk∩h= 0.

∙ Γ is a uniform lattice in G acting freely on G/K.

The Lie group H associated with h acts (locally free) on the quotient Γ∖G/K by rightmultiplication.


A related, but weaker, notion is the notion of quasi-algebraic action.

Definition 3.3.2. A quasi-algebraic (nilpotent) action is given by a quadruple (G,K,Γ,a), where

∙ G is a connected Lie group with Lie algebra g

∙ K is a closed subgroup of G with Lie algebra k

∙ a is a nilpotent subalgebra of g, contained in the normalizer Ngk of k and such thatk∩a= 0.

∙ Γ is a discrete subgroup G acting freely on G/K such that Γ∖G/K is compact.

The Lie group H associated with a acts (locally free) on the quotient Γ∖G/K by rightmultiplication.

Lemma 3.3.3. (BARBOT; MAQUERA, 2013) An algebraic action (G,K,Γ,h) is Anosov, if,and only if, there exists an element h ∈ h and an ad(h) invariant splitting g= U ⊕S ⊕h⊕ k

such that the eigenvalues of ad(h)|U (resp. ad(h)|S ) have strictly positive (resp. negative) realpart.

We shall consider SO(k,k+n) as the matrix subgroup of GL(R2k+n) of elements thatpreserves the bilinear form

⟨u,v⟩=k

∑i=1

uivi −2k+n

∑i=k+1

uivi.

Using this identification, we can consider the Lie algebra so(k,k+n) also as a matrixLie algebra given by

so(k,k+n) =

so(k) C X t

Ct so(k) Zt

X Z D

; C ∈ Mk×k; X ,Z ∈ Mn×k; D ∈ so(n)

(3.5)

We consider I ⊂ Mk×k the set of diagonal matrices and embed I on so(k,k+n) in anatural way, that is

I =

0 J 0

J 0 00 0 0

∈ so(k,k+n) ; J ∈ Mk×k is diagonal

(3.6)

It is clear that I induces an action of Rk on SO(k,k+n). Moreover, as this action leavesSO(n) invariant, it descends to an action on SO(k,k+n)/SO(k).


3.3.1.1 The action is Anosov

Proposition 3.3.4. Let Γ be an uniform lattice in SO(k,k+ n), acting freely in SO(k,k+ n)/

SO(k). If J = diag(λ1, . . . ,λk) ∈ I is such that 0 = λi =±λ j for all i = j, then J is an Anosovelement of the action (by right multiplication) I yΓ∖SO(k,k+n)/SO(k)

Proof. The main tool of the proof is Lemma 3.3.3. It is quite clear that the action is algebraic.It remains to study the adjoint action of I on g= so(k,k+n). We will show that there existsa I invariant splitting, g = I ⊕F ⊕ so(n) such that for a J ∈ I under the conditions of theproposition, it’s eigenvalues are real and non zero on F . By grouping together the eigenspacescorresponding to the positive and negative eigenvalues we obtain the desired F = U ⊕S

splitting.

Let J ∈ I and

M =

R1 C X t

Ct R2 Zt

X −Z D

∈ so(k,k+n)

and we set

F =

M ∈ so(k,k+n) ; D = 0 ; C =

0 * * . . . ** 0 * . . . *... . . . ...* . . . * * 0

then:

[J,M] =

JCt −CJ JR2 −R1J JX t

JR1 −R2J JC−CtJ JZt

XJ −ZJ 0

We want to consider the eigenvalues of this action, more specifically, we want to show

that for any J = diag(λ1, . . . ,λk) such that 0 = λi =±λ j for all i = j there exists a J invariantsplitting g = I ⊕S ⊕U ⊕ so(n) where the action of J on S (resp U ) has positive (respnegative) eigenvalues. Now consider J = diag(λ1, . . . ,λk). First observe that J acts on the X andZ part by multiplication, and the eigenvalue problem reduces to JX = αX (similarly for Z). Thus,each element λ j corresponds to an eigenvalue with n dimensional eigenspace. Now suppose thatR2 = 0, X = 0 Z = 0, we obtain

[J,M] =

JCt −CJ −R1J 0JR1 JC−CtJ 0

0 0 0


From the eigenvalue problem [J,M] = αM we obtain

0 = JC−CtJ (3.7)

R1 = α(JCt −CJ) (3.8)

C =−αR1J (3.9)

After a little manipulation we obtain

R1 = α2(J2R1 +R1J2)

Now write Ei j, 1 ≤ i < j ≤ k for the generators of so(k) that contains a 1 in the i, j

position and a −1 on the j, i position. Taking R1 = Ei j, then

α2(J2R1 +R1J2) = (λ 2

i −λ2j )Ei j

Thus, α±i j =

±1√λ 2

i −λ 2j

are eigenvalues with corresponding eigenvectors given by solving

3.7 above. Moreover, from the hypothesis λi = ±λ j = 0, these eigenvalues are real and nonzero. Doing a similar calculation for R1 = 0, X = 0, Z = 0 we obtain the other eigenvectors.Dimensional analysis show that this is in fact a complete set of eigenvectors17. Now it is justa matter of grouping our eigenvectors in positive and negative associated eigenvalues. Thecorresponding spanned spaces U and S satisfies the hypothesis of Lemma 3.3.3.

3.3.1.2 The action is generalized k-contact

Our goal in this section is to show, that the action we considers comes from a generalizedk-contact action.

The general strategy for the proof used in this section will be used latter (on section3.3.4) to generalize this example. Roughly speaking, the idea is to choose 1-forms on I andextend those to so(k,k+n) by using the splitting so(k,k+n) = I ⊕S ⊕U ⊕ so(n) obtainedin the previous section. Afterwards we show that if we avoid certain hyperplanes (finitely many),the chosen 1-forms will define the generalized k-contact structure.

Remember that a k-contact structure on M is a collection of 1-forms satisfying certainproperties (Definition 3.2.1). To compensate our lack of information about Γ we shall defineleft invariant one forms on SO(k,k+n)/SO(n). The left invariance means that those forms willdescend to 1-forms on Γ∖SO(k,k+n)/SO(n).

Now, our goal is to define some specific left invariant one forms on SO(k,k+n)/SO(n)

and make some computations. If we denote by Ωq(G/K,R)G the space of G-invariant q-forms17 By a complete set of eigenvector, we mean that they generate F , that is if we sum the corresponding

eigenspaces with I and so(n) we recuperate the whole g.


on G/K and Hom(Λqg/k,R)K the space of K-invariant, q-multi-linear, alternating maps on g/k,then, it is known that

Ωq(G/K,R)G = Hom(Λqg/k,R)K

We shall work first on Λqg*.

First, we stablish some notations and do some preliminary computations. We denote byCi j ∈ Mk×k the matrix whose only non zero coordinate is the i, j coordinate, whose value is 1.As before, we denote Ei j =Ci j −C ji.

From Ci jCkl = δ jkCil we obtain

Ei jCkl = δ jkCil −δikC jl (3.10)

CklEi j = δilCk j −δ jlCki (3.11)

Now, consider I ⊂ Mk×k the subspace of diagonal matrices, and embed it into so(k,k+

n):

I =

0 J 0

J 0 00 0 0

∈ so(k,k+n) ; J is diagonal

We take α ∈ I * and we consider α as a linear form on so(k,k+n) by extending it to

zero according to the obvious basis. By definition, this linear form is zero on so(n) and alsoSO(n)-invariant18, thus, it defines a left invariant 1-form on SO(k,k+n) which descends to a1-form on SO(k,k+n)/SO(n).

We shall make our calculations on SO(k,k+n). For left invariant vector fields A,B, wehave that α(A) and α(B) are constants, and thus the usual formula

dα(A,B) = A(α(B))−B(α(A))−α([A,B])

reduces to dα(A,B) =−α([A,B]). Thus, for the purpose of our calculations, we are interestedonly in the diagonal portion of the Mk×k on so(k,k+n), that is, for two given matrices M1,M2 ∈so(k,k+n), in the expression of

[M1,M2] =

R1 C X t

Ct R2 Zt

X Z D

,R1,R2 ∈ so(k),D ∈ so(n),X ,Z ∈ Mn×k,C ∈ Mk×k

18 Just notice that I is in the normalizer of so(n) and I ∩ so(n) = 0


we are only interested in the matrix C. With this notation, we shall write

JM1,M2K =C

Moreover, as we are actually interested only in the diagonal elements of C, we shalldenote

Diag

c11 c12 . . . c1n

c21 c22 . . . c2n...

... . . . ...cn1 cn2 . . . cnn

=

c11 0 . . . 00 c22 . . . 0...

... . . . ...0 0 . . . cnn

=

c11

c22...

cnn

and define

|JM1,M2K|= DiagJM1,M2K

If we write

M1 =

R1 A X t

At R2 Zt

X Z D

and M2 =

R1 A X t

At R2 Zt

X Z D

then

JM1,M2K = R1A+AR2 +X t Z − R1A− AR2 − X tZ

In what follows, we will make frequent use of the different injections Mk×k,so(k),Mn×k →so(k,k+n), and thus, we fix the following notation:

For any 1 ≤ i ≤ j ≤ k we will denote

Fi j =

Ei j 0 00 0 00 0 0

; Gi j =

0 0 00 Ei j 00 0 0

; Hi j =

0 Ci j 0C ji 0 00 0 0

We will also make no distinction between X ,Z ∈ Mn×k and their images: 0 0 X t

0 0 0X 0 0

and

0 0 00 0 Zt

0 −Z 0

We shall denote by Xi j,Zi j ∈ Mn×k the matrix with 1 in the i, j coordinate and zero in all

others.

It is clear that for s, t = i, j we have

|JFi j,HstK|= |JGi j,HstK|= 0


then, from 3.12,3.15 and 3.16 it follows that dαkn+2k(k−1) = 0 and moreover, ker(dα) = so(n)⊕I .

Finally, as α and its differential dα vanish everywhere on so(n) and are SO(n) invariant,they correspond to forms on the quotient SO(k,k+n)/SO(n) and our calculations remain valid,that is, dα has constant rank 2kn+2k(k−1)+1. Choosing a basis α1, . . . ,αk of the dual spaceI * satisfying the condition 3.22, we obtain that for every j ∈ 1, . . . ,k

α1 ∧·· ·∧αk ∧dαkn+k(k−1)j is a volume form on SO(k,k+n)/SO(n)

Therefore, such 1-forms α1, . . . ,αk defines a generalized k-contact structure on SO(k,k+

n)/SO(n). Moreover, by definition, the generalized k contact action is the action of the Cartansubspace I .

3.3.1.3 It is not k-contact

The goal of this section is to prove that the action of the Cartan subspace we have beenconsidering in the previous two sections is not associated with a k-contact structure (Definition3.2.27). More precisely, we will prove the following proposition

Proposition 3.3.5. Consider M = SO(k,k+n)/SO(n), k > 1, let I ⊂ so(k,k+n) be the abeliansubalgebra given by 3.6 and E = U ⊕S ⊂ so(k,k+n) (Proof of Proposition 3.7). Let I and E

be the corresponding left SO(k,k+n)-invariant distributions on M. Then, there does not existsleft invariant 1-forms α1, . . . ,αk on M such that these forms define a k-contact structure on M

and such thatk⋂

j=1

kerα j = E and I = kerdα j 1 ≤ j ≤ k

Proof. As before (3.5) we consider so(k,k+n) as matrix Lie algebra

so(k,k+n) =

so(k) C X t

Ct so(k) Zt

X Z D

; C ∈ Mk×k; X ,Z ∈ Mn×k; D ∈ so(n)

(3.21)

and

I =

R1 C X t

Ct R2 Zt

X Z D

∈ so(k,k+n) ; R1 = R2 = 0 ; X = Z = 0 ; D = 0 ; C is diagonal

We also have Hi j ∈ so(k,k+n) defined by

Hi j =

0 Ci j 0C ji 0 00 0 0


where Ci j ∈C ∈ Mk×k is the matrix with value 1 on the coordinate (i, j) and value 0 otherwise.

The condition

∙ α1 ∧·· ·∧αk ∧dαni is a volume form on M

on the definition of k-contact, means that dαi must be non degenerate on E = ∩kerα j. Fromprevious considerations, we know that left invariant 1-forms on M can be identified with SO(n)-invariant linear forms on so(k,k+n)/so(n). The additional condition α|E = 0 implies that α canin fact, be identified with an element of I *. In particular, the values (and certain properties) ofdα depends only on the values of α on I 19, more precisely, it depends on the values of α onthe following subspace of I

|Jso(k,k+n),so(k,k+n)K| ⊂ I = SpanRH11, . . . ,Hkk

However, the computations on the previous chapter show us that

|Jso(k,k+n),so(k,k+n)K|= SpanRH11, . . . ,Hkk

Explicitly,|JFi j,Hi jK|=−H j j

Thus, suppose that α1,α2 are two of the 1-forms that define the k-contact structure. Wehave dα1 = dα2 if, and only if α1(H j j) = α2(H j j), ∀ j = 1, . . . ,k. However, H11, . . . ,Hkk is infact a basis of I , and therefore,

dα1 = dα2 ⇔ α1 = α2

which contradicts the hypothesis that α1,α2 are linearly independent.

Remark 3.3.6. For the case k = 1 the Cartan action is in fact the geodesic flow on a negativelycurved symmetric space (Katok, A. and Hasselblatt, B (KATOK; HASSELBLATT, 1997)), it istherefore a contact flow20. Moreover, if k = 1, both the k-contact structure and the generalizedk-contact structure reduces to the usual contact structure. This explain why the condition k > 1on the proposition is necessary.19 In fact, most of the work in the previous section was to prove that dα is non degenerate on E if, and

only if

0 = α(H j j) =±α(Hii) ∀i = j, (3.22)

20 It is a well established (if somewhat folkloric) fact of the contact geometry that the unit tangent bundlehas canonical contact structure and it’s Reeb flow gives the geodesic flow (For a complete proof in theslightly more general context of Finsler geodesic see Dorner, M. and Geiges, H. and Zehmisch, K..(DöRNER; GEIGES; ZEHMISCH, 2016)).


3.3.1.4 For k = 2 it is not a contact pair

The goal of this section is to prove that, for k = 2 the action of the Cartan subspace wehave been considering is not associated with a contact pair (Definition 3.2.23). More precisely,we will prove the following proposition

Proposition 3.3.7. Consider M = SO(2,2 + n)/SO(n), let I ⊂ so(2,2 + n) be the abeliansubalgebra given by 3.6 and E = U ⊕S ⊂ so(k,k+n) (Proof of Proposition 3.7). Let I and E

be the corresponding left SO(2,2+n)-invariant distributions on M. Then, there does not existsleft invariant 1-forms α,η on M such that these forms define a contact pair on M where at leastone Reeb vector field is Anosov, and such that

kerα ∩kerη = E and I = kerdα ∩kerdη

Proof. As before, we consider so(2,2+n) as matrix Lie algebra

so(2,2+n) =

0 r1

−r1 0a b

c dX t

a c

b d

0 r2

−r2 0Zt

X Z D

; a,b,c,d,r1,r2 ∈ R;X ,Z ∈ Mn×2; D ∈ so(n)

(3.23)

and

I =

0 00 0

a 00 d

0

a 00 d

0 00 0

0

0 0 0

∈ so(2,2+n) ; a,d ∈ R

As we have previously seen, left invariant 1-forms on M that satisfies the desired con-

ditions correspond to elements in I *, therefore, the condition α , η are linearly independent21

means that the choice of α and η corresponds (by duality) to a choice of basis for I .

We shall take a basis Xα ,Xη defined as

Xα =

0 00 0

1 00 a

0

1 00 a

0 00 0

0

0 0 0

Xη =

0 00 0

d 00 1

0

d 00 1

0 00 0

0

0 0 0

21 The hypothesis that α ∧η ∧dαh ∧dηk is a volume form implies that α and η are linearly independent

(otherwise α ∧η = 0.)


where ad = 1. And let α,η be the corresponding dual basis.

From the Section 3.3.1.1, the Anosov condition means that either a ∈ 0,±1 or d ∈0,±1.

Our goal is to prove that for any choice of constants a,d ∈ R∖±1 such that ad = 1,the pair (α,η) is not a contact pair.

As Xα ,Xη is not the canonical basis of I , it is convenient to see how the canonicalbasis is written in terms of Xα ,Xη .

Let J ∈ I written as

J =

0 00 0

S 00 T

0

S 00 T

0 00 0

0

0 0 0

S,T ∈ R

then

J = λαXα +ληXη

where

λα = λα(S,T ) =S−dT1−ad

; λη = λη(S,T ) =T −aS1−ad

If we use the notations of Section 3.3.1.2, we can write

J = SH11 +T H22

and

λα(S,T ) = Sα(H11)+T α(H22)

λη(S,T ) = Sη(H11)+T η(H22)

In particular,

α(H22) = λα(0,1) =−d

1−ad

η(H11) = λη(1,0) =−a

1−adand

α(H11) =1

1−ad= η(H22)

From the computations of Section 3.3.1.2 we have


dα2n(X11,Z11, . . . ,X2n,Z2n)

= Π1≤s≤nα(H11)α(H22) =(−dn)

(1−ad)2n = 0

and similarly

dη2n(X11,Z11, . . . ,X2n,Z2n) =

(−an)

(1−ad)2n = 0

This shows that the rank of α and η must be at least 2n, and if (α,η) is a contact pair oforder (h,k), then, by dimensional analysis, we have three distinct possibilities:

∙ (h,k) = (n+2,n)

∙ (h,k) = (n+1,n+1)

∙ (h,k) = (n+2,n+2)

Now, still using the notation from Section 3.3.1.2, so(k)⊕ so(k) is generated by thematrices Fi j,Gi j1≤i< j≤k, as k = 2 we set F = F12 and G = G12.

From previous computations

dα(F,H21) = α(H11) =1

1−ad= 0

dη(G,H21) =−η(H11) =− 11−ad

= 0

Which implies that h ≥ n+1 and k ≥ n+1. However we also have

dα2(F,G,H12,H21) = 2(α(H11)

2 −α(H22)2) =

2(1+d)(1−d)(1−ad)2

and

dη2(F,G,H12,H21) =

2(1+a)(1−a)(1−ad)2

If d = ±1 we conclude that dα has rank n+ 2, and if a = ±1 we conclude that dη

has rank n+2. The dimensional analysis implies that we must have d,a ∈ +1,−1. But thiscontradicts the Anosov condition, which finishes the proof.


3.3.2 Geometric interpretation of SO(k,k+n)/SO(n)

In this subsection, we shall revisit the example in the previous section with a moregeometric flavour. After setting some definitions and basic results we define an action whichwe prove to be Anosov. To do so we define an left invariant metric and make some explicitcalculations. Afterwards, we define a family of 1-forms and prove the necessary conditions toshow that they define a generalized k-contact Anosov structure.

Remark 3.3.8. In what follows, small Latin letters x,y,z,a,b,c will denote vectors (with dimen-sions to be specified), with coordinates xi,yi,zi,ai,bi,ci, respectively.

3.3.3 Definitions

Consider on Rk,k+n the 2k+n-dimensional real vector space endowed with a non degen-erate bilinear form of signature (k,k+n), explicitly, let x,y,a,b ∈ Rk and c,z ∈ Rn, then,

(a,b,c) · (x,y,z) =k

∑i=1

aiyi +bixi +n

∑j=1

c jz j

Lemma 3.3.9. We can identify SO(k,k+ n)/SO(n) with the set of 2k- vectors (up,vp), up =

(xp,yp,zp), vp = (ap,bp,cp), p = 1, . . . ,k, such that

1. up ·uq = 0 = vp · vq

2. up · vq = δpq

3. dim(Span⟨u1,v1, · · · ,uk,vk⟩) = 2k

With this identification, we will denote an element of SO(k,k+ n)/SO(n) by the pair (u,v),where u = (u1, . . . ,uk) and v = (v1, · · · ,vk)

Proof. If we denote by M the set defined above, then, SO(k,k+n) acts on M transitively, andthe isotropy subgroup is precisely SO(n).

Lemma 3.3.10. Using the above identification, we can also identify T(u,v)M with the set of 2k-vectors (up, vp), 1 ≤ p ≤ k, such that

1. up · uq +uq · up = 0

2. vp · vq + vq · vp = 0

3. up · vq + vq · up = 0

As before, we denote an element of T(u,v)M by the pair (u, v), where u = (u1, · · · , uk) andv = (v1, · · · , vk).


Proof. We just differentiate the defining equations of M. The third condition that defines M isnot relevant for it is an open condition.

3.3.3.1 Anosovness

For each (u,v)∈M we consider the decomposition Rk,k+n =V(u,v)⊕W(u,v)⊕Z(u,v) where

V(u,v) = Span⟨u1, . . . ,uk⟩

W(u,v) = Span⟨v1, . . . ,vk⟩

Z(u,v) = (V(u,v)⊕W(u,v))⊥

Now, we write for, (u, v) ∈ T(u,v)M the vectors up and vp in this basis:

up =k

∑i=1

µpiui +νpivi +wp (3.24)

vp =k

∑i=1

µ′piu

i +ν′piv

i +wp′ (3.25)

(3.26)

wp,wp′ ∈ Z.

From conditions (3.3.9) we have

up · vq = ∑i

µpiui · vq = ∑i

µpiδiq = µpq

up ·uq = νpq

vp ·uq = ν′pq

vp · vq = µ′pq

Lemma 3.3.11. For (u, v) ∈ T(u,v)M, the formula

‖(u, v)‖2 = ∑p≥q

|µpq|2 + |νpq|2 + |µpq′|2 + |ν ′pq|2 +∑

q‖wq‖2 +‖wq′‖2

defines an SO(k,k+n) invariant metric on M.

Proof. Let us first check that this is in fact a norm, that is,

‖(u, v)‖= 0 ⇒ (u, v) = 0

But ‖(u, v)‖2 = 0 implies that for every p ≥ q

|µpq|= |νpq|= |µpq′|= |ν ′pq|= ‖wq‖= ‖wq′‖= 0


From 3.3.10, for every p,q

0 =up · vq + vq · up = ν′qp +µpq

0 =up · uq +uq · up = νqp +νpq

0 =vp · vq + vq · vp = µ′qp +µ

′pq

And thus, µpq,µ′pq,νpq,ν

′pq are zero for any p,q, as we wanted.

Let us check that this is SO(k,k+n) invariant. First, notice that SO(k,k+n) preservesthe splitting Z ⊕Z⊥, and moreover, it acts on Z by orthogonal transformations. Finally, noticethat it also preserves all the products22, and thus, it preserves the quantities µpq,µ ′

pq,νpq and ν ′pq

for they are given by products.

It is straightforward to show that this norm actually satisfies the parallelogram law, andthus, comes from a inner product, and thus define a metric on M.

Remark 3.3.12. Notice that we have also prove that the coordinates µpq,µ ′pq,νpq and ν ′

pq arecompletely determined if we know just their values for p ≥ q.

Theorem 3.3.13. For a given λ = (λ1, . . . ,λk)∈Rk, and a collection of k vectors u= (u1, . . . ,uk)

we denote eλ u = (eλ1u1, . . . ,eλkuk). Then, the action of Rk on M given by

φ : Rk ×M → M

(λ ,(u,v)) ↦→ φλ (u,v) = (eλ u,e−λ v)

is Anosov.

Lemma 3.3.14. If φ is the action above, then dφλ acts on the tangent space by

(dφλ )(u,v)(u, v) = (eλ u,e−λ v)

In particular, if we write

up =k

∑i=1

µpiui +νpivi +wp

and

eλ up =k

∑i=1

µλpiu

i +νλpivi +wλ p

then wλ p = eλ wp, and similarly for vp

22 By definition, SO(k,k+n) is the set of transformations that preserves the products.


Proof. Notice that we can understand our manifold M as embedded in the linear space (Rk,k+n)2k,and in this case, the action is actually linear, therefore, its differential is itself.

Proof of Theorem 3.3.13. For λ = (λ1, . . . ,λk) and (u, v) ∈ T(u,v)M we have

‖dφtλ (u, v)‖2 = ∑p≥q

e(λp−λq)t |µpq|2 + e(λp+λq)t |νpq|2 + e−(λp−λq)t |µ ′pq|2 + e−(λp+λq)t |ν ′

pq|2

+∑q

e2λqt‖wq‖2 + e−2λqt‖wq ′‖2

We shall suppose that λp > λq > 0 if p > q.

We define

S :=(u, v) ; µpq = νpq = 0 ∀p ≥ q ; µ′pp = wp = 0 ∀p

U :=(u, v) ; µ′pq = ν

′pq = 0 ∀p ≥ q ; µpp = wp′ = 0 ∀p

andλ+0 = max

p≥q(λp −λq),(λp +λq),(2λp)

λ−0 = min

p≥q(λp −λq),(λp +λq),(2λp)

Then, for t > 0

‖(dφtλ )|S (u, v)‖2 = ∑p≥q

e−(λp−λq)t(1−δpq)|µ ′pq|2 + e−(λp+λq)t |ν ′

pq|2 +∑q

e−2λqt‖wq′‖2

≤ eλ+0 t

∑p≥q

µ′pq|2 +ν

′pq|2 +∑

q‖wq′‖2 ≤ eλ

+0 t‖(u, v)‖2

and analogously,‖(dφ−tλ )|U (u, v)‖2 ≤ e−λ0t‖(u, v)‖2

That is, the distribution U (resp. S ) is expanded (resp. contracted) uniformly.

Define also

I := (u, v) ; µpq = ν′pq = 0 ∀p > q

νpq = µpq′ = 0 ∀p ≥ q

wp′ = wp = 0 ∀p

that is, I is the set of (u, v) ∈ T(u,v)M such that, in the basis (3.24), we have

up = µppup

vp = ν′ppvp

The condition up · vp + vp · up = 0 implies µpp =−ν ′pp, and thus

I = (u, v) ∈ T(u,v)M ; (up, vp) = (λpup,−λpvp) for some λ1, . . . ,λk ∈ R


that is

I = (u, v) ∈ T(u,v)M ; (u, v) = λ (u,v) for some λ ∈ Rk

To see that this distribution is tangent to the action, take a curve γ = (u(t),v(t)) =

(eλ tu0,e−λ tv0). It is clear that (u(t), v(t)) = (λu(t),−λv(t)) is the field of tangent vectors alongγ .

It remains to show that T M = U ⊕I ⊕U . This is the content of the following lemma

Lemma 3.3.15. Let S , U and I defined as in the previous proof. Then

T M = U ⊕I ⊕U

Proof. Let (u, v) ∈ T(u,v)M, and let us write

up = ∑j

µp ju j +νp jvi +w j

= µppup︸︷︷︸upI

+ ∑j>p

µp ju j

︸︷︷︸upS

+j

∑j<p

µp ju j +∑j

νp jv j +νppvp +wp

︸︷︷︸upU

vp = ∑j

µ′p ju

j +ν′p jv

j +wp

= ν′ppvp︸︷︷︸vpI

+∑j

µ′p ju

j + ∑j<p

ν′p jv

j +wp′

︸︷︷︸upS

+j

∑j>p

νp jv j

︸︷︷︸upU

It is a straightforward calculation to show that (uS , vS ), (uI , vI ) and (uU , vU ) aretangent to M. Moreover,

(u, v) = (uS , vS )+(uI , vI )+(uU , vU )

and

(uS , vS ) ∈ U

(uI , vI ) ∈ I

(uU , vU ) ∈ V

Remark 3.3.16. In the proof of Theorem 3.3.13 above, we have assumed λp > λq > 0 if p > q

and we have shown that such λ = (λ1, . . . ,λk) is an Anosov element. We do not need this


condition. We only must suppose that λp = λq = 0 for p = q, but the study of the signs inthis general case becomes cumbersome. That is, for the different choices of λ , we must definedifferent U =Uλ and S =Sλ . However, whatever the case may be, the direct sum distributionSλ ⊕Uλ is the same.

3.3.3.2 Generalized k-contact

Consider, for j = 1, . . . ,k the 1-forms η j defined by

(η j)(u,v)(u, v) = u j · v j

Clearly η j is SO(k,k+n)-invariant.

Lemma 3.3.17. S ⊕U =⋂k

j=1 kerη j

Proof. First, we observe that

k⋂j=1

kerη j = (u, v) ; up · vp = µpp = ν′pp = 0 ∀p

and thus

S ⊕U ⊂k⋂

j=1

kerη j

Now, we take (u, v) ∈⋂k

j=1 kerη j and write its decomposition T M = I ⊕U ⊕ V

according to Lemma 3.3.15.

(u, v) = (uS , vS )+(uI , vI )+(uU , vU )

By definition,

(upI , vp

I ) = (µppup,ν ′ppvp),

and thus, (uI , vI ) = 0, that is (u, v) ∈ U ⊕V

Let us write up = (xp,yp,zp), vp = (ap,bp,cp), p = 1, . . . ,k, then, in this coordinates wehave

ηp = up · vp =k

∑i=1

xpi dbp

i + ypi dap

i +n

∑l=1

zpl dcp

l

and

dηp =k

∑i=1

dxpi ∧dbp

i +dypi ∧dap

i +n

∑l=1

dzpl ∧dcp

l

Lemma 3.3.18. For any (u, v) ∈ I , we have i(u,v)dηp = 0


Proof. First, we observe that we are seeing the form dηp as a 2-form on the linear space(Rk,k+n)2k and (Rk,k+n)2k with itself. Thus, we shall write

(u, v) =p

∑j=1

up∂up + vp

∂vp

where

up∂up =

k

∑l=1

xpl ∂xp

l+ yp

l ∂ypl+

n

∑l=1

zpl ∂zp

l

and analogous for vp.

We have seen that

(u, v) ∈ I ⇔∃(λ1, . . . ,λk) ∈ Rk ; (up, vp) = (λpup,−λpvp) ∀p

And thus,

(u, v) =k

∑j=1

λpup∂up −λpvp

∂vp

It is clear thati∂uqdηp = i∂vqdηp = 0 for q = p

therefore

i(u,v)dηp = iλpup∂updηp − iλpvp∂vpdηp

λp = λp

( k

∑i=1

xpi ∧dbp

i + ypi ∧dap

i +n

∑l=1

zpl ∧dcp

l

)−λp

( k

∑i=1

−dxpi ∧bp

i −dypi ∧ap

i +n

∑l=1

−dzpl ∧ cp

l

)= λp

( k

∑i=1

xpi dbp

i + ypi dap

i +bpi dxp

i +api dyp

i +n

∑l=1

zpl dcp

l + cpl dzp

l

)That is,

i(u,v)dηp(∂xql) = λbq

l

i(u,v)dηp(∂yql) = λaq

l

i(u,v)dηp(∂zql) = λcq

l

and analogous for the tangent vectors ∂aql, ∂bq

land ∂cq

l.

Thus, for a general tangent vector (u, v) = ∑kj=1 up∂up + vp∂vp we have

i(u,v)dηp((u, v)) = λp

( k

∑i=1

xpi bp

i + ypi ap

i +bpi xp

i +api yp

i +n

∑l=1

zpl cp

l + cpl zp

l

)= λp(up · vp + vp · up)


But, if (u, v) ∈ T(u,v)M then up · vp + vp · up = 0 and thus i(u,v)dηp((u, v)) as we wanted.

Remark 3.3.19. From the previous computations, it is clear that for two vectors (u, v), (u, v) ∈T(u,v)M we have

dη j((u, v),(u, v)) = u j · v j − v j · u j

Remark 3.3.20. Consider an invertible matrix B = (βi j) ∈ Mk×k. Then, the 1-forms

αi := ∑j

βi jη j

satisfiesk⋂

j=1

kerα j =k⋂

j=1

kerη j

and moreover,

i(u,v)dα j = 0

Theorem 3.3.21. For almost every coordinate change B, the 1-forms αi = ∑ j βi jη j define ageneralized k-structure on M.

Proof. We consider a change off coordinates B = (βi j) such that βis =±βit = 0 for every s, t.

We will show that for every i, dαi is non degenerate on U ⊕S . To avoid cluttering thenotation, we suppress the index i, that is we consider a linear combination α = ∑ j β jη j such thatβs =±βt = 0 for every j. We want to show that dα is non degenerate on U ⊕S . As each η j

is SO(k,k+n)-invariant, then so is dα and as SO(k,k+n) acts transitively on M, then we onlyhave to study dα on a single point.

Take the point (u,v) given by

up = (ep, ~Ok, ~On) ∈ Rk,k+n and vp = (~0k,ep,~0n) ∈ Rk,k+n

where ep denotes the p’th element of the canonical basis e1, . . . ,ek of Rk, and~0d denotes thezero vector of Rd . Clearly (u,v) ∈ M.

On this point, if (u, v) has coordinates

up = (xp, yp, zp)

vp = (ap, bp, cp)

then we can write the conditions 3.3.10 as

1. yip + yp

i = 0

2. aip + ap

i = 0


3. xip + bp

i = 0

In particular

ypp = ap

p = 0 and xpp +bp

p = 0 (3.27)

That, is, if we restrict dα = ∑kp,i=1 βpdxp

i ∧dbpi +dyp

i ∧dapi +∑

nl=1 dzp

l ∧dcpl to T(u,v)M then we

can use the identities

1. dyip +dyp

i = 0

2. daip +dap

i = 0

3. dxip +dbp

i = 0

in particular, dypp = 0, dap

p = 0 and dxpp ∧dbp

p = 0 and we obtain

dα =k

∑p,i=1

βp(dxp

i ∧dbpi +dyp

i ∧dapi +

n

∑l=1

dzpl ∧dcp

l

)= ∑

l,pβpdzp

l ∧dcpl +(∑

p>i+∑

p<i+∑

p=i)βpdxp

i ∧dbpi +βpdyp

i ∧dapi

= ∑l,p

dzpl ∧dcp

l + ∑p>i

(βp −βi)dxpi ∧dbp

i +(βp +βi)dypi ∧dap

i

It is clear that for βp =±β i = 0 we have

ker(dα|T(u,v)M) = (u, v); up = (t1ep, t2ep,~0n) ; vp = (t3ep, t4ep,~0n)

where t1, t2, t3 and t4 are arbitrary values constrained by the equations (u, v) ∈ T(u,v)M. That is,t2 = t3 = 0 and t1 + t4 = 0.

This means that, over T(u,v)M, ker(dα) is precisely the points of the form (λu,−λv),which we have seen is the subspace I tangent to the orbit of the action φ .

We conclude that dα is non degenerate on T M/I = S ⊕U

3.3.4 Algebraic actions

The goal of this section is to generalize the construction in the previous section and proveTheorem 3.3.26 below. Before we state the theorem, let us recall some definitions and resultsabout Lie algebras.

Definition 3.3.22. Consider a Lie algebra g. A Cartan subalgebra c of g is an abelian subalgebra,such that ad(x) is semisimple23 for every x ∈ c, and maximal for these properties.23 Recall that an endomorphism T of a (finite dimensional) vector space V is called simple if any invariant

subspace is trivial, and it is called semisimple if it is a sum T1 ⊕·· ·⊕Tl ∈ End(V1 ⊕·· ·⊕Vl) of simpleendomorphisms Tj ∈ End(Vl)


Definition 3.3.23. Consider a Lie algebra g. A Cartan subspace a of g is an abelian subalgebra,such that, for every x ∈ a, the linear map ad(x) is hyperbolic (its eigenvalues are all real andpositive), and maximal for these properties.

Lemma 3.3.24. If a is a Cartan subspace of g, then it’s centralizer Zg(a) can be written as

Zg(a) = k⊕a

where k is the Lie algebra of a compact subgroup.

Lemma 3.3.25. If g0 is a real Lie algebra and h0 is a Cartan subalgebra of g0, then, thecomplexification h= h0

C = h⊕ ih of h0 is a Cartan subalgebra of g= g0C (the complexification

of g0).

We are ready to enunciate the main theorem of this section.

Theorem 3.3.26. Consider a real, connected, semisimple, non compact, Lie group G, with finitecenter and with Lie algebra g. Let a be the Cartan subspace and K ⊂ G the compact groupassociated with the compact part of the center of a. Consider also a uniform lattice Γ in G actingfreely on G/K. The action of the Cartan subspace a on Γ∖G/K, is an generalized k-contactAnosov action.

Before we prove this theorem, we give some definitions and previous known results thatwill be used in our proof.

Proposition 3.3.27. (HOF, 1985) Consider G a non compact semisimple real Lie group withfinite center and Lie algebra g. Consider a torsion free, uniform lattice, Γ in G and a Cartansubspace a of G. Let a⊕ k be the centralizer of a in g, and K ⊂ G the compact subgroup withLie algebra k (see Lemma 3.3.24). The algebraic action (G,K,Γ,a) is always Anosov.

Remark 3.3.28. By the Proposition 3.3.27, the action in Theorem 3.3.26 is Anosov. It remainsto show that this action is associated with a generalized k-contact structure. The main tool is astructure theorem of real semisimple Lie algebras as exposed by Holger Kammeyer ((KAM-MEYER, 2014)). In what follows we develop the necessary notations to enunciate Kammeyer’stheorem.

Definition 3.3.29. Let g be a Lie algebra over a field F. The Killing form B on g0 is the bilinearform

B : g×g→ F

(x,y) ↦→ B(x,y) := Tr(ad(x)∘ad(y))

A Cartan involution of g, is a involution map θ : g→ g such that the bilinear map

Bθ : g×g→ F

(x,y) ↦→ Bθ (x,y) := B(x,θy)


is positive definite.

Remark 3.3.30. If g is a semisimple Lie algebra and F has characteristic 0, then the Killingform is non degenerate and a Cartan involution always exists. The (generalized) eigenspaces k, passociated with the eigenvalues +1 and −1 of θ , gives a θ -invariant decomposition

g= k⊕p

called the Cartan decomposition of g.

Remark 3.3.31. If g0 is a real Lie algebra and θ is a Cartan involution on g0, there exists aunique C-linear extension θC of θ to the complexification g = g0 ⊕ ig0 of g0. We shall abusenotation and denote θC = θ .

Lemma 3.3.32 ((HELGASON, 1979),(KAMMEYER, 2014)). Consider g0 a real semisimpleLie algebra, a Cartan involution θ on g0, and the corresponding Cartan decomposition g0 = k⊕p.

There exists a maximal abelian, θ -stable subalgebra h0 ⊆ g0 such that a0 = h0 ∩p is amaximal abelian subalgebra in p.

Lemma 3.3.33 (Helgason,(HELGASON, 1979), Chapter XI). Under the above notations, h0 isa Cartan subalgebra of g0 and a is a Cartan subspace24 of g0.

Definition 3.3.34. Let g be a reductive Lie algebra over a field F of characteristic zero. A Cartansubalgebra h is called a splitting for g if adg(x) is triangularizable over F for every x ∈ h. Wecall the pair (h,g) a F-splitting Lie algebra, or simply a splitting Lie algebra.

Lemma 3.3.35. If g is a Lie algebra over an algebraically closed field F, then every Cartansubalgebra c is a splitting for g.

Definition 3.3.36. Consider a splitting Lie algebra (h,g), and denote h* the dual vector space ofh. For λ ∈ h* and h⊂ E ⊂ g an ad(h)-invariant vector space, we denote

Eλ = Eλ (h) := Y ∈ E ; (ad(Z)−λ (Z)Id)nY = 0 ; for some n ∈ N ∀Z ∈ h

and we define∆(h,E) = λ ∈ h*0 ; Eλ = 0

The elements of ∆(h,g) are called roots of (h,g) and the decomposition

g= h⊕⊕

λ∈∆(h,g)

gλ

is called the root space decomposition.24 Helgason does not call this space a Cartan subspace. However, he shows that for any x ∈ a, ad(x) is

hyperbolic. Moreover, if g= k⊕a⊕n denotes the Iwasawa decomposition, and g = k+a+n is thedecomposition of g ∈ g, then we can write ad(g) = KAN, where K is elliptic, A is hyperbolic and N isunipotent. Moreover, K = Id (resp N = Id) if and only if k = 0 (resp n = 0). Therefore, a is maximaland thus a Cartan subspace of g0.


Remark 3.3.37. If (h,g) is a F-splitting Lie algebra, where F is algebraically closed, then Eλ isat most one dimensional and

EλY ∈ E ; ad(Z)Y = λ (Z)Y ; ∀Z ∈ h

Definition 3.3.38. The set Y ∈ h ; λ (Y ) = 0 ∀λ ∈ ∆(h,g) has a finite number of connectedcomponents called Weyl chambers. For a fixed choice of Weyl chamber W , we can define

∆+(h,g) = ∆

+(h,g,W ) = λ ∈ ∆(h,g) ; λ (x)> 0 ∀x ∈ W

The roots on ∆+(h,g) are called positive roots, with respect to W , and we speak about"choice of positivity", meaning that a Weyl chamber was chosen and the positive roots are thosethat are positive with respect to this Weyl chamber.

Definition 3.3.39. A positive root is called simple if it is not a positive linear combination ofother positive roots. The set of simple positive roots will be denoted by ∆

+S (h,g).

Consider a real semisimple Lie algebra g0 with a Cartan involution θ . Consider theCartan subalgebra h0 obtained in Lemma 3.3.32. Let g= g0

C the complexification of g0, then,h= h0

C ( the complexification of h0) is a Cartan subalgebra of g and therefore (h,g) is a splittingLie algebra.

We let ∆+S (h,g) = α1, . . . ,αl be a choice of simple positive roots.

Denote by σ the anti-linear complex automorphism of g given by the conjugation withrespect ot g0, that is,

σ : g= g0 ⊕ ig0 → g0 ⊕ ig0

g = g1 + ig2 ↦→ σ(g) = g1 − ig2

For each α ∈ ∆(h,g) we define

ασ (h) = α(σ(h)).

Notice that if xα ∈ gα , then

[h,σ(xα)] = σ([σ(h),xα ]) = σ(α(σ(h))xα) = α(σ(h))σ(xα)

and therefore (σ(xα)) ∈ gασ

, that is ασ is also a root.

We also denote by ∆R the space of real roots (roots fixed by σ ), ∆iR the space ofimaginary roots (roots fixed by the Cartan involution θ , and therefore vanishes on a) and ∆C thecomplex roots (neither real or imaginary). We set ∆

+*C be ∆

+C with one element removed from

each pair α,ασ. We also denote ∆*C = ∆

+*C ∪−∆

+*C .


Theorem 3.3.40 (Kammeyer, (KAMMEYER, 2014), Theorem 4.1). There exists a partition∆(h,g) = ∆1∪∆0 such that ∆(h,g)∩∆iR ⊂ ∆0 and H 1 = H1

α ; α ∈ ∆1 and H 0 = H0α ; α ∈

∆0 are basis of a and h0 ∩ k and a basis B of g0 given by

B := X0α ,X

1α ; α ∈ ∆

+iR∪∆

*C∪Zα ; α ∈ ∆R∪H 1 ∪H 2

such that,

1. [H iα ,H

jβ] = 0 for every α,β .

2. [H iα ,X

jβ] = ci j

αβX i+ j+1

β.

3. [H iα ,Zβ ] = [H i

α ,Xsgn(β )−1

2β

].

4. For β ∈ −α,−ασ; [X iα ,X

jβ] = (−1)i jγαβ X i+ j

α+β+ sgn(α)γασ β X i+ j

ασ+β.

5. For α ∈ ∆*C; [X i

α ,Xj−α ] = (−1)i jH1+i+ j

α where 2H0α and H1

α are non zero Z-linear combi-nations in H 0 and H 1 respectively.

6. For α ∈ ∆+iR; [X0

α ,X1α ] = H0

α , where H0α is a nonzero Z linear combination of elements H0

β,

where β ∈ ∆(h,g)∩∆iR ⊂ ∆0.

7. [Zα ,Z−α ] =−sgn(α)H1α where 2H1

α is a non zero Z-linear combination in H 1.

8. [Zα ,X iβ] = [X

sgn(α)−12

α ,X iβ].

and the constants ci jαβ

and γαβ are half integers and satisfy.

1. γα,β = γ−α,−β

2. γαβ =±(r+1) where r is the largest integer such that β − rα ∈ ∆(h,g)

3. ci jαβ

=−ci, j+1αβ

4. ci jα,β = (−1)i+1ci j

α,β σ

5. c1 jα,β = 0 if β ∈ ∆iR.

Remark 3.3.41. On items 5 and 6 on Theorem 3.3.40 above, we have imposed that H 0, H 1,H 0

α and H 1α are non zero. These conditions were not stated explicitly on Kammeyer’s paper

((KAMMEYER, 2014)). They are, however, a clear consequence of the proof. Those vectors areobtained by constructive methods, and a careful analysis of the construction show them to benon zero.

We are now ready to prove Theorem 3.3.26


Proof of Theorem 3.3.26. The idea of the proof is to imitate the construction in section 3.3. Moreprecisely, we shall take a splitting g0 = a⊕ (k∩h0)⊕N +⊕N − where N ± are the nilpotentparts of opposing parabolic subalgebras.

Using the idea of section 3.3, we shall take 1-forms on a* and using this splitting, extendit to 1-forms on g0 and, like in section 3.3, we shall prove that if we chose 1-forms η1, . . . ,ηk

that are non zero on certain vectors, then, they define the desired generalized k-contact structure.

As we have already remarked (Remark 3.3.28), under the conditions of our theorem, theaction is Anosov, moreover, almost25 every element of a is Anosov.

Thus it remains to choose the 1-forms

Let us denoteN ± = SpanRX i

α ,Zβ ,α ∈ ∆*±C ,β ∈ ∆

±R

,K = SpanRX0

α ,X1α ; α ∈ ∆

+iR⊕SpanRH 0

andH = SpanRH 1

We have the decomposition

g0 = K ⊕H ⊕N +⊕N − (3.28)

where H is a Cartan subspace and K is a compact subalgebra. Take λ ∈ H *. We can use thedecomposition (3.28) to extend λ to a linear functional on g0, and therefore we can understand itas a left invariant 1-form on G. Moreover, as K is in the centre of H , than λ is K-invariant and,therefore, can be seen as a left invariant 1-form on G/K. Because it is left invariant, it pass on tothe quotient Γ∖G/K.

We want conditions to show that its differential dλ is non degenerate on N +⊕N −.We just have to choose λ that does not vanishes on H1

α , H1α

26. For such λ , we obtain that foreach element Y ∈ X i

α ,Zβ ,α ∈ ∆*±C ,β ∈ ∆

±R basis of N +⊕N −, there exists Y * in the same

basis such that dλ (Y,Y *) =−λ ([Y,Y *]) = 0 as we wanted. Moreover, the conditions on λ areopen, and we can choose a basis of such λ satisfying the condition. It is clear that this basis willdefine the generalized r-contact structure on G/K.

25 Here, almost every element means that the complement is a finite union of hyperplanes.26 Such λ does exists, for H1

α , H1α are always non zero and are finite in number.

85

CHAPTER

4MAIN THEOREM

The goal of this section is to prove the following theorem

Theorem 4.0.1. Let (M2n+k,α,φ) be a generalized k-contact Anosov action. Then, it is smoothlyconjugated to a quasi algebraic action.

The strategy of our proof is to endow our manifold with a (G′,G′/H ′)-structure that iscompatible with the Anosov action. This compatibility means that it give us the desired smoothconjugacy, and moreover, the induced action on G′/H ′ is of algebraic nature, that is, it is theright action of an abelian subalgebra. The proof can be outlined in the following steps.

Step 1: Define a rigid Gromov geometric structure compatible with our given structure anddynamics. In particular, Rk acts on M by automorphisms of this structure. As the Anosovaction is transitive, the open-dense orbit theorem will ensure the existence of an opendense subset Ω ⊂ M which is locally homogeneous. This is done in Subsection 4.1.

Step 2: Fix a point with compact orbit in Ω. Let H ′ be the group of local automorphisms thatstabilizes this point. Denote its Lie algebra by h′. Let g′ be the Lie algebra of Killing vectorfields of the geometric structure around this point. Let G′ be the associated connectedsimply connected Lie group, and H ′ the connected subgroup with Lie algebra h′. Our goalis to show that Ω can be locally modelled after G′/H ′. This is done in the Subsection4.2.4. The second (and most technical) step of the proof is to show that H ′ is closed inG′ (Proposition 4.2.36). This requires some technical preparation which is done over theSubsections 4.2.1,4.2.2 and 4.2.3.

Step 3: G′/H ′ is a model space for Ω. The third step (Section 4.3) is to show that we can extendthis structure, that is Ω = M.

86 Chapter 4. Main Theorem

Step 4: The last step of the Part 1 is to show that the geometric structure we defined (now in thesense of Ehreshman-Thurston) is complete, that is, the developing map is a covering map(Proposition 4.4.1).

4.1 The geometric structure

We shall now, define a rigid geometric structure associated with our generalized k contactAnosov action. Let m = 2n+ k and V = Rm. Denote by Grn(V ) the Grassmanian of n-planes inV and define Σ as

Σ :=(A1, . . . ,Ak,e+,e−,ω1, . . . ,ωk), with

A j ∈V ∀ j e± ∈ Grn(V ) such that

V = e+⊕RA1 ⊕·· ·⊕RAk ⊕ e−

ω j ∈ Λ2(V *) such that, for each j ∈ 1, . . . ,k

(ω j)|e+⊕e−is non degenerated

kerω j = RA1 ⊕·· ·⊕RAk

It is clear that Σ is algebraic and have a natural algebraic action of GL(m,R). Now wedefine an A-structure of order 1 and type Σ on M by:

σ : F1(M)→ Σ

(p,ξ ) ↦→ ξ−1((X1)p, . . . ,(Xk)p,(E+)p,(E−)p,(dα1)p, . . . ,(dαk)p)

Remark 4.1.1. Notice that our geometric structure also encodes the data of the 1-forms definingthe geometric structure, for given a point in Σ, there exists unique linear functionals α1, . . . , αk

on V such that αi(A j) = δi j and e+⊕ e− ⊂ ker(αi). However, the geometric structure deal withinfinitesimal data, and thus cannot encode how such linear functionals should relate with thebilinear forms ω1, . . . ,ωk.

Notice that the Killing vector fields of this geometric structure, are precisely the vectorfields Y such that for every j = 1, . . . ,k

[Y,X j] = 0

[Y,E±]⊂ E±

LY α j = 0

4.1. The geometric structure 87

Moreover, notice that for each j, we can define a pseudo-Riemannian metric g j on M by

g j(Xi,Xl) = δil

g j(Xi,Z+) = g j(Xi,Z−) = 0

g j(W+,Z+) = g j(W−,Z−) = 0

g j(Z+,W−) = dα j(Z+,W−)

for all 1 ≤ i, l ≤ k, Z+,W+ ∈ Γ(M,E+), Z−,W− ∈ Γ(M,E−).

Proposition 4.1.2. The geometric structure σ is rigid.

Remark 4.1.3. Essentially, the rigidity of the geometric structure σ follows from the fact that itsautomorphisms actually preserves a pseudo-Riemannian metric, which are rigid. In what followswe formalize this intuition.

Proof. Define Σ0 as the following

Σ0 :=(A1, . . . ,Ak,e+,e−,ω), with

A j ∈V ∀ j e± ∈ Grn(V ) such that

V = e+⊕RA1 ⊕·· ·⊕RAk ⊕ e−

ω ∈ Λ2(V *) such that

ω|e+⊕e−is non degenerate

kerω = RA1 ⊕·· ·⊕RAk

and for each j, define the geometric A-structure σ j of order 1 and type Σ0

σ j : F1(M)→ Σ0

(p,ξ ) ↦→ ξ−1((X1)p, . . . ,(Xk)p,(E+)p,(E−)p,(dα j)p)

Suppose that each σ j is rigid. This means that form some l ≥ 1 we have

Aut l+1pp (σ j)

π lj→ Aut l

pp(σ j)

is injective for all j. It is clear that

Aut lpp(σ) =

⋂j

Aut lpp(σ j).

Thus the map ⋂j

Aut l+1pp (σ j)→

⋂j

Aut lpp(σ j)

is injective and thus σ is rigid.

The following lemma proves that σ j is rigid.


Lemma 4.1.4. For j = 1,2, we have the inclusion Aut jpp(σ j) → Aut j

pp(g j), for every p ∈ M,where g j is understood as an A-structure and σ j is 1-rigid.

Proof. The inclusion is quite clear. To avoid cluttering the notation, we shall denote σ = σ j andg = g j To understand that σ is rigid, we consider Σσ and Σg, the Σ’s of the geometric structuresσ and g respectively. Suppose that there exists a map ρ : Σσ → Σg (GL(Rm)-invariant) such thatρ ∘σ = g. And suppose that g is r-rigid. Then, consider φ ∈ Autk(σ) We have the followingdiagram

Fk(M)

Fk(M)

Σσ

Σg

φ ρg

σ

g

It is clear that φ is also an automorphism of g, moreover, if ψ is an automorphism of gl,

the diagram above induces a diagram

Fk+l(M)

Fk+l(M)

Σlσ

Σlg

ψ ρgl

σ l

gl

We obtained the following situation

· · ·

· · ·

Autr+1(σ)

Autr+1(g)

Autr(σ)

Autr(g)

· · ·

· · ·

pr+1k (σ)

pr+1k (g)

As both the vertical arrows and pr+1k (g) are injective, then, so are pr+l

k (σ), as we wanted.

It remains to construct the map ρ:

(A1, . . . ,Ak,e+,e−,λ ) ↦→

λ 0 00 1 00 0 1

where we understand λ as a bilinear form with matrix λ and use the identification Rm =

e+⊕ e−⊕RA1 ⊕·· ·⊕RAk.

4.1. The geometric structure 89

Remark 4.1.5. The previous lemma generalizes in the following way. Consider two geometricstructures on Mm σ1 and σ2 of types Σ1 and Σ2 and orders r1 and r2 respectively. Suppose thatfor some r ≥ 0 and s1,s2 ≥ 0 such that r1 + s1 = r2 + s2 = r there exists a Glr(m)-equivariantmap

ρ : Σs11 → Σ

s22

such that ρ ∘σs11 = σ

s22 . Then, if σ2 is rigid, so is σ1.

Corollary 4.1.6. The pseudo-group of local diffeomorphisms of M that preserves the distribu-tions E± and X j and the 1-forms α j have a dense orbit Ω.

Proof. This follows from the topological transitivity of the action and Gromov’s open denseorbit theorem..

Definition 4.1.7. The open and dense set Ω ⊂ M where the pseudo-group Autloc(σ) acts transi-tively will be called the domain of the geometric structure.

Now, for a given Anosov element X with flow ϕt , we consider the strong stable andunstable leaves through p, F±

X (p).

F±(p) = F±X (p) = q ∈ M| lim

t→∓∞d(ϕt(q),ϕt(p)) = 0

where d is the distance function associated with the riemmanian metric.

Lemma 4.1.8. For a given Anosov element X , consider the set

∆ = ∆X = p ∈ M|F+X (p)⊂ Ω and F−(p)⊂ Ω

Then, ∆ is dense in M.

Proof. First, we notice that compact orbits are dense in M (they are dense in the non wanderingset, which we have proved to be all M). As Ω is open and dense, it follows that compact orbitsare dense in the entire Ω.

We shall prove that ∆ contains every compact orbit in Ω. It is known that the Anosovfoliations depends only on the open connected cone of Anosov elements we chose. Let p ∈ Ω

with a compact orbit. We can take an Anosov element X ′ with flow ϕ ′t in the same cone as X

such that ϕ ′t0(p) = p.

Now, let q ∈ F+(p), then,

0 = limt→+∞

d(ϕ ′t (q),ϕ

′t (p)) = lim

n→∓∞d(ϕ ′

nt0(q),ϕ′nt0(p))

= limn→∓∞

d(ϕ ′nt0(q), p)


thus ϕ ′nt0(q) → p. As Ω is open, and invariant by the action, it follows that q ∈ Ω. We have

proved that F+(p)⊂ Ω. A similar argument proves the case F−(p).

4.2 The model space

Fix a point v0 ∈ Ω with compact orbit, and let g′ be the set of germs of Killing vectorfields at v0. Let h′ be the Lie algebra of the isotropy group H ′ of v0. Let G′ be the connected,simply connected Lie group associated to g′ and H ′ be the connected subgroup of G′ associatedwith h′. Our goal in this section is to prove that H ′ is closed in G′. To do this, we must studythe Lie algebra g′ in great detail. Our main tool for this study will come from a collection ofconnections ∇ j on T M which are "compatible" with the geometric structure.

On Subsection 4.2.1, we build this connections and give some of their properties. Theseconnections will induce connections the determinant bundle ΛE± (bundle of volume forms onE±) (Subsection 4.2.2). This connections will turn out to be flat (Theorem 4.2.8, Definition 4.2.9Theorem 4.2.10, Theorem 4.2.23)

On Subsection 4.2.3, we consider lie algebra g′ and using parallel sections of the determi-nant bundle ΛE±, we construct g the semisimple part of the Levy decomposition of g′ (Lemma5.1.1).

On subsection 4.2.4 we finally show that H ′ is closed in G′ (Proposition 4.2.36) andconstruct a model for the open dense subset Ω (Lemma 4.2.44).

4.2.1 An adapted connection

In this section we define a collection of affine connections on M which are compatiblewith our geometric structure, that is, for which the Killing vector fields are infinitesimal affinetransformations. This will allow us to use Lemma 2.3.15 to better understand our geometricstructure. We also obtain an explicit formula for the torsion, and obtain some properties forthe curvature, which we shall use later to prove that the Lie algebra of Killing vector fields isreductive. Our definition of the connection is inspired by the definition given at (BENOIST;FOULON; LABOURIE, 1992).

Consider a k-contact Anosov action on M.

Lemma 4.2.1. For every j ∈ 1, . . . ,k, and real valued linear functionals C±j : Rk → R there

exists unique smooth connection ∇ j on M that satisfies

∇jdα j = 0 ; ∇

jαi = 0 ∀i ; ∇

j(E±)⊂ E± ; ∇jT φ ⊂ T φ (4.1)

4.2. The model space 91

and for X ∈ Γ(M,T φ), Z± ∈ Γ(M,E±)

∇jZ+Z− = p−([Z+,Z−]) (4.2)

∇jZ−Z+ = p+([Z−,Z+]) (4.3)

∇jX Z± = [X ,Z±]+C±

j (X)Z± (4.4)

Where p± denotes the projection T M → E± and we use the natural identification Rk = Tpφ =

RX1(p)⊕·· ·⊕RXk(p) to compute C±(X).

Proof. Notice that a connection can be seen as a bilinear map

∇ : Γ(M,T M)×Γ(M,T M)→ Γ(M,T M)

which satisfies a Leibniz condition

∇X( fY ) = X( f )Y + f ∇XY ; ∀ f ∈C∞(M,R)

and is C∞(M,R)-linear in the first coordinate. The smooth splitting T M = E+⊕RX1 ⊕ ·· ·⊕RXk ⊕E− induces a splitting

Γ(M,T M) = Γ(M,E+⊕)Γ(M,RX1)⊕·· ·⊕Γ(M,RXk)⊕Γ(M,E−)

and bi-linearity means that we can define our connection on each component of this splitting.

Notice that the conditions (4.2),(4.3),(4.4) defines a bilinear map on appropriate subspacesof Γ(M,T M)×Γ(M,T M), which satisfies the Leibniz condition and is C∞(M,R)-linear in thefirst coordinate, it remains to define ∇ on Γ(M,E+)×Γ(M,E+), Γ(M,E−)×Γ(M,E−) and onΓ(M,T M)×Γ(M,RXi), that is, we must specify, for Z±,G± ∈ Γ(M,E±) and G ∈ Γ(M,T M),the vector fields

∇jG+Z+ ; ∇

jG−Z− and ∇

jGX j

We actually will see that the conditions (4.1) - (4.4) forces what must be these vectorfields, and therefore the obtained connection is unique.

First, we shall prove that a connection ∇ j wich satisfies the hypothesis of the Lemma4.2.1, also satisfies ∇ jXi = 0 for every 1 ≤ i ≤ k. This is of course bilinear and satisfies theLeibniz condition. For the sake of clearer notation, we shall avoid using the j superscript on theconnection, and simply denote ∇ instead of ∇ j.

From ∇αi = 0 we have, for any vector fields Y,Z,

0 = (∇Y αi)(Xs) = Y (αi(Xs))−αi(∇Y Xs) =−αi(∇Y Xs)


and from ∇dα j = 0 and the fact that Xs ∈ ker(dα j) we obtain

0 = ((∇Y dα j)(Xs,Z)) = Y (dα j(Xs,Z))−dα j(∇Y Xs,Z)−dα j(Xs,∇Y Z)

=−dα j(∇Y Xs,Z) =−(i∇Y Xsdα j)(Z)

Thus, i∇Y Xsdα j = 0 and i∇Y Xsαi = 0 therefore i∇Y XsdM j = 0. But dM j is a volume form and, thus,∇Y Xs = 0.

Now, we just have to define ∇G±Z±. For this, we notice that ∇dα j = 0 means that forevery vector field A,B,C we have

0 = (∇Cdα j)(A,B) = LC(dα j(A,B))+dα j(∇CA,B)−dα j(A,∇CB)

But dα j restricted to E+⊕E− is non degenerate, thus we define ∇G±Z± to be the unique vectorfield tangent to E± which satisfies.

dα j(∇G±Z±,Y∓) = LG±(dα j(Z±,Y∓))−dα j(Z±,∇G±Y∓)

This is of course bilinear and C∞(M,R)-linear on the G± coordinate. It remains to seethat it satisfies the Leibniz condition: Let f be a smooth function on M, then by definition

dα j(∇G± f Z±,Y∓) = LG±(dα j( f Z±,Y∓))−dα j( f Z±,∇G±Y∓)

= LG±( f dα j(Z±,Y∓))− f dα j(Z±,∇G±Y∓)

= LG±( f )(dα j(Z±,Y∓))+ f LG±(dα j(Z±,Y∓))− f dα j(Z±,∇G±Y∓)

= (dα j(LG±( f )Z±,Y∓))+ f dα j(∇G±Z±,Y∓)

= (∇ f G±Z±+LG±( f )Z±,Y∓)

and thus ∇G± f Z± = ∇ f G±Z±+LG±( f )Z± as wanted.

Remark 4.2.2. Notice that, by construction, for all j, the connection ∇ j defined above iscompatible with our geometric structure, that is, ∇ j is invariant by Autloc(σ), moreover, thesmoothness of the invariant bundles means that this connection is smooth.

Lemma 4.2.3. Let T j be the torsion of the connection ∇ j above, denote by C±i j =C±

j (Xi) andSi j =C+

i j p++C−i j p−. Then,

T j(A,B) = ∑i

(αi([p+B, p−A]+ [p−B, p+A])Xi +αi(A)Si jB−αi(B)Si jA

)= ∑

idαi(A,B)Xi +αi(A)Si jB−αi(B)Si jA

)

and ∇ jT j = 0. Moreover, if K j denotes the curvature of ∇ j, then for any A,Z ∈ Γ(M,E+) andB ∈ Γ(M,E−) we have:

K j(A,B)Z = 0


Proof. The first step of the proof is to compute the torsion, by the bi-linearity of the torsiontensor, we can decompose the tangent space and compute the torsion on each pair of components.

To show that ∇ jT j = 0 we show, using some straightforward computations, that ∇jZT j(A,B)=

∑ciXi for some functions ci = ci(A,B,Z) and then, we use the Bianchi identity to conclude that∇

jZT j(A,B) must be tangent to E+⊕E−. The Bianchi Identity will also be used to compute

K j(A,B)Z for A,Z ∈ Γ(M,E+) and B ∈ Γ(M,E−).

Let’s compute the torsion. In what follows, we shall omit the index j and write ∇ = ∇ j

T = T j and C± =C±j

First, note that for X tangent to T φ and Z = Z± ∈ Γ(M,E±)

T (X ,Z) = ∇X Z −∇ZX − [X ,Z]

= [X ,Z]+C±(X)Z − [X ,Z] =C±(X)Z

Moreover, for given 2−form ω and vector fields G,Z,Y we have

iG ∘ iZ ∘ iY dω = ω([Z,Y ],G)+ω(Z, [G,Y ])+LY (ω(Z,G))−LZ(ω(Y,G))+

+ω(Y, [Z,G])+d(ω(Y,Z))(G)

So, if we take ω = dα j and Y = Y± and Z = Z±, we know that dα j(Y,Z) = 0 andddα j = 0, that is

0 = dα j([Z±,Y±],G)+dα j(Z±, [G,Y±])+LY±(dα j(Z±,G))−LZ±(dα j(Y±,G))+

+dα j(Y±, [Z±,G])

If we take G = G∓ we obtain

dα j(T (Y±,Z±),G∓) = 0

On the other hand,

T (Y±,Z±) = ∇Y±Z±−∇Z±Y±+[Z±,Y±]

and thus, T (Y±,Z±)⊂ E± and therefore dα j(T (Y±,Z±),G∓) = 0 implies

T (Y±,Z±) = 0.

Now, if we write Id = p++ p−+∑i αi( · )Xi then


T (Y±,Z∓) = p∓([Y±,Z∓])− p±([Z∓,Y±])− [Y±,Z∓]

= (p±+ p∓− Id)([Y±,Z∓]) =−∑i

αi([Y±,Z∓])X j

= ∑j

dαi(Y±,Z∓)X j

If we write A = Z++X +Z− and B =W++Y +W−, then dαi(A,B) = dαi(Z+,W−)+

dαi(Z−,W+) and we obtain from the bi-linearity of T

T (A,B) = T (Z+,W−)+T (Z−,W+)+T (X ,W++W−)−T (Y,Z++Z−)

= ∑i

(dαi(Z+,W−)+dαi(Z−,W+)

)Xi +C+(X)W++C−(X)W−−C+(Y )Z+−C−(Y )Z−

If we write C±i j =C±

j (Xi) then C(X) =C±j (X) = ∑i αi(X)C±

i j

If we write Si j =C+i j p++C−

i j p−

T j(A,B) = ∑i

dαi(A,B)Xi +αi(A)Si jB−αi(B)Si jA

Now, let’s prove that ∇T j = 0.

As before, we continue to omit the index j. In particular, we write Si for Si j

From ∇Xi = 0 we have ∇Z( f Xi) = LZ( f )Xi. Moreover, the projections p± commuteswith ∇ and thus:

∇Z(T (A,B)) = ∑i

(LZ(dαi(A,B))X j +LZ(αi(A))SiB+αi(A)Si∇ZB−

−LZ(αi(B))SiA+αi(B)Si∇ZA)

On the other hand, we have

0 = (∇Zαi)(A) = LZ(αi(A))−αi(∇ZA)

so

∇Z(T (A,B)) = ∑i

(LZ(dαi(A,B))X j +αi(∇ZA)SiB+αi(A)Si∇ZB−

−αi(∇ZB)SiA+αi(B)Si∇ZA)


Thus,

∇ZT (A,B)−T (∇ZA,B)−T (A,∇ZB) = (4.5)

= ∑i

(LZ(dαi(A,B))−dαi(∇ZA,B)−dαi(A,∇ZB)

)Xi (4.6)

= ∑i

((∇Zdαi)(A,B)

)Xi (4.7)

Notice that if either A or B belongs to the central direction, T φ the above formula is zeroas we wanted. The formula also vanish if both A and B belongs to the (un)stable distribution.From symmetry, we can consider A ∈ Γ(M,E+) and B ∈ Γ(M,E−).

First we shall consider Z = Xs for some s ∈ 1, . . . ,k.

In this case dαi(∇ZA,B) =−αi[[Z,A]+C+(Z)A,B] and dαi(A,∇ZB) =−αi[A, [Z,B]+

C−(Z)B] and thus, as C++C− = 0 we obtain

∇ZT (A,B)−T (∇ZA,B)−T (A,∇ZB) =

=−∑i

(LZ(αi([A,B]))−αi([Z,A],B]+ [A, [Z,B]])

)Xi

=−∑i

(LZ(αi([A,B]))−αi([Z, [A,B]])

)Xi

=−∑i

(LZ(αi([A,B]))− [A,B](αi(Z))−αi([[A,B],Z])

)Xi

=−∑i

dαi(Z, [A,B])Xi = 0

Now we consider Z = Z+ ∈ Γ(M,E+). Let U be a set and E a vector space. For any mapΞ : U ×U ×U → E, we denote the cyclic sum over a,b,c ∈U as:

S(Ξ(a,b,c)) := Ξ(a,b,c)+Ξ(b,c,a)+Ξ(c,a,b)

using such notation, the well known Bianchi formula can be written as

S(K(A,B)Z) =S(T (T (A,B),Z))+∇ZT (A,B)

where K denotes the curvature of the connection, that is

K(A,B) = ∇A∇B −∇B∇A −∇[A,B]

Now, for Z,A tangent to E+ we have T (A,Z) = 0, and for B ∈ Γ(M,E−) we obtain

S(T (T (A,B),Z) = T (T (A,B),Z)+T (T (B,Z),A)

= ∑i

dαi(A,B)T (Xi,Z)+∑i

dαi(B,Z)T (Xi,A)

= ∑i

dαi(A,B)C+i Z +dαi(B,Z)C+

i A


and thus S(T (T (A,B),Z) is tangent to E+.

As ∇E± ⊂ E± it follows that

S(K(A,B)Z) ∈ Γ(M,E+⊕E−)

As ∇ZT (A,B) = ∑i((∇Zdαi)(A,B)

)Xi it follows that ∇ZT (A,B) = 0.

Moreover, from

S(K(A,B)Z) =S(T (T (A,B),Z)) ∈ Γ(M,E+)

andS(K(A,B)Z) = K(A,B)Z︸︷︷︸

∈Γ(M,E+)

+K(B,Z)A︸︷︷︸∈Γ(M,E+)

+K(Z,A)B︸︷︷︸∈Γ(M,E−)

it follows that K(Z,A)B = 0

Corollary 4.2.4. For any i ∈ 1, . . . ,k we have ∇ jdαi = 0

Proof. It follows from the previous proof that

(∇ZT )(A,B) = ∑i((∇Zdαi)(A,B))Xi.

As ∇T = 0 it follows that (∇Zdαi)(A,B) = 0

Lemma 4.2.5. If K is the curvature tensor of ∇ then K(E±,E±) = 0

Proof. Let ω be any 2-form. Denote by H(A,D) = (∇Bω)(C,D). Then

(∇A∇Bω)(C,D) = (∇AH)(C,D) = AH(C,D)−K(∇AC,D)−H(C,∇AD)

= A(Bω(C,D)−ω(∇BC,D)−ω(C,∇BD)

)−(Bω(∇AC,D)−ω(∇B∇AC,D)−ω(∇AC,∇BD)

−(Bω(C,∇AD)−ω(∇BC,∇AD)−ω(C,∇B∇AD)

and

(∇[A,B])ω(C,D) = L[A,B](ω(C,D))−ω(∇[A,B]C,D)−ω(C,∇[A,B]D)

thus

(K(A,B)ω)(C,D) = ω(K(B,A)C,D)+ω(C,K(B,A)D)

in particular, for ω = dαi, as ∇dαi = 0 (and therefore K(A,B)dαi = 0), we have:

0 = dαi(K(B,A)C,D)+dαi(C,K(B,A)D)


From the previous lemma, K(Y±,Z±)G∓ = 0 for any Y±,Z± tangent to E± and G∓

tangent to E∓, and therefore, for any W± tangent to E±,

dλ (K(Y±,Z±)W±,G∓) = dλ (K(Y±,Z±)G∓,W±) = 0

and thus, K(Y±,Z±)W±= 0. Evidently K(Y±,Z±)X j = 0 and we can conclude that K(Y±,Z±)=

0.

Lemma 4.2.6. Consider the curvature K as a 2-form taking values on End(R2n+ k). As theconnection preserves the splitting T M = E+⊕T φ ⊕T−, we can consider the restriction of this2-form to the sub-bundles E±. In this case we have

K(Xl,Y )|E± = 0 ∀l ∈ 1, . . . ,k, Y ∈ T M

Proof. Once again we use the linearity of the curvature tensor on each coordinate to facilitateour computations. This allow us to break the proof in in three cases:

Case 1: Y = Xi

Case 1: Y = Xi

Case 2: Y = Y∓

Case 3: Y = Y±

We consider a section Z± of E±. Then,

∇Xl ∇XiZ± = ∇Xl([Xi,Z±]+C±

i Z±)

= [Xi, [Xl,Z±]]+C±l [Xi,Z±]+C±

i [Xl,Z±]+C±i C±

l Z±

and thus, from Jacobi Identity, it follows that

K(Xl,Xi)Z± = [Xi, [Xl,Z±]]− [Xl, [Xi,Z±]] = [Z±, [Xl,Xi]] = 0

Case 2: Y = Y∓

We have

∇Xl ∇Y∓Z± = ∇Xl(p±[Y∓,Z±])

= [Xl, p±[Y∓,Z±]]+C±l p±[Y∓,Z±]


and

∇Y∓∇Xl Z± = ∇Y∓([Xl,Z±]+C±

l Z±)

= p±[Y∓, [Xl,Z±]]+C±l p±[Y∓,Z±]

Thus

K(Xl,Y∓)Z± = ∇Xl ∇Y∓Z±−∇Y∓∇Xl Z±−∇[Xl ,Y∓]Z

±

= [Xl, p±[Y∓,Z±]]− p±[Y∓, [Xl,Z±]]− p±[[Xl,Y∓],Z±]

+C±l p±[Y∓,Z±]−C±

l p±[Y∓,Z±]

= [Xl, p±[Y∓,Z±]]− p±[Y∓, [Xl,Z±]]− p±[[Xl,Y∓],Z±]

Now, from the integrability, [Xl, p±A] = p±[Xl,A], and therefore, we write

K(Xl,Y∓)Z± = p±([Xl, [Y∓,Z±]]− [Y∓, [Xl,Z±]]− [[Xl,Y∓],Z±)

From Jacobi’s identity, it follows that K(Xl,Y∓)Z± = 0

Case 3: Y = Y±

Lets denote ? := dα j(K(Xl,Y±)Z±,Z∓). It is clear that K(Xl,Y±)Z± is completelydetermined by ?. To simplify, in what follows, we denote X = Xl , Y = Y± Z+ = Z± andZ− = Z∓.

From ∇dα j = 0 we have

dα j(∇X ∇Y Z+,Z−) = X(dα j(∇Y Z+,Z−))−dα j(∇Y Z+,∇X Z−)

= X(dα j(∇Y Z+,Z−))−(Y (dα j(Z+,∇X Z−))−dα j(Z+,∇Y ∇X Z−)

)and thus

?=dα j(∇X ∇Y Z+,Z−)−dα j(∇Y ∇X Z+,Z−)−dα j(∇[X ,Y ]Z+,Z−)

=X(dα j(∇Y Z+,Z−))−Y (dα j(Z+,∇X Z−))+dα j(Z+,∇Y ∇X Z−)−(Y (dα j(∇X Z+,Z−))−X(dα j(Z+,∇Y Z−))−dα j(Z+,∇X ∇Y Z−)

)−(

[X ,Y ](dα j(Z+,Z−))−dα j(Z+,∇[X ,Y ]Z−))

=X(dα j(∇Y Z+,Z−)+dα j(Z+,∇Y Z−)

)−

Y(dα j(∇X Z+,Z−)+dα j(Z+,∇X Z−)

)+

[X ,Y ](dα j(Z+,Z−))+dα j(Z−,∇Y ∇X Z−−∇X ∇Y Z−−∇[Y,X ]Z−)

From the previous case, we know that the last term is zero, and from

Y (dα j(Z+,Z−)) = dα j(∇Y Z+,Z−)+dα j(Z+,∇Y Z−)


we obtain

?= X(Y (dα j(Z+,Z−)))−Y (X(dα j(Z+,Z−)))− [X ,Y ](dα j(Z+,Z−)) = 0

which finishes the proof.

Lemma 4.2.7. The geodesics of ∇ which are tangent to the unstable (stable) distribution Euu

(Ess) are complete.

Proof. Associated with the connection ∇ we have the concepts of exponential map and normalneighbourhoods, which are analogous to the Riemannian case. On such a neighbourhood, theexponential map is give us the geodesic. Since M is compact, there exists a constant c > 0 suchthat for every tangent vector Y of M with ‖Y‖ < c we can integrate the geodesic with initialcondition Y to a time equal or greater the one.

Now, suppose that we have Y ∈ Euu, this means that there exists t0 such that for t < t0we have

‖dφ(ta, · )Y‖< c

and through dφ(ta, · )Y we can integrate the geodesic to a time equal or greater then one. Butthe action φ is affine (with respect to ∇), which means it transport geodesics, and therefore, wecan integrate the geodesic through Y to a time equal or greater then one. As Y is arbitrary, thegeodesics tangent to Euu are complete.

4.2.2 Bundle of volume forms

We now study the determinant bundle of E±, i.e the bundle Λ± of volume forms onE±, that is, we consider the vector bundle E± and take the exterior n-power Λn(E±)*, wheren = dimE±. It is easy to see that this bundle is of rank 1.

This bundle can be endowed with a connection (∇ j)′, induced from the connection ∇ j

on E±. We shall abuse notation and write (∇ j)′ = ∇ j.

Our goal is to prove

Theorem 4.2.8. The bundle Λ± endowed with the connection ∇ j has curvature tensor Ω j,±

identically zero.

A consequence of this theorem is that the associated bundle, Λ±, on the universal coverM of M, also has zero curvature and, therefore, there exists global parallel sections, unique up to


a constant. Those parallel sections will be of great importance for our study of the Lie algebra g′.

In what follows, we shall (usually) drop the superscript ±, and whenever it is needed weshall choose the stable bundle E+, but know that all the constructions done can be done for boththe stable and the unstable bundle.

The curvature Ω j = Ω j,± on this bundle is given1 by

Ωj,±(X ,Y ) = Tr(K(X ,Y )|E± )

Definition 4.2.9. Observe that Θ defines a bilinear, non degenerate, skew symmetric form onE+⊕E−. Therefore, we can define an operator B j = B j,± on E+⊕E− by

Ωj(Y,Z) = dα j(B jY,Z) (4.8)

We can further extend B j by setting

αi(B jZ) = 0 ∀i

Our strategy to prove that (Λ+,∇ j) is flat, is to show that B j is zero. To see this, we willshow that B j is both nilpotent and semisimple.

Our goal for this section is to prove

Theorem 4.2.10. The operator B j is nilpotent

Before proving this, lets build a primitive of Ω j. Let ζ = ζ j be a section of Λ+. If Λ+ istrivial, we can assume this section to be never vanishing, otherwise, we take a section ζ modulosign (that is, a section of |Λ+|2).

Now, remember that the connection form β of a vector bundle E (with respect to a localframe s) is given by

∇Es = s ·β

where β can be understood as a End(E) valued form. On the case at hand, our bundle is uni-dimensional, and our local frame is just a single section ζ . We define thus the (real valued) oneform β = β j by

∇jZζ

j = βj(Z)ζ j

1 Check the Appendix B, equation (B.4)2 The fiber of |Λ+| is actually R/±Id= [0,+∞)


Finally, observe that the curvature form Ω of E (with respect to the local frame s) isgiven (in terms of the connection form β ), by:

Ω = dβ +β ∧β

On the case at hand, β is actually a "normal" real valued one form3, and therefore,β ∧β = 0, thus,

Ωj = dβ

j

Lemma 4.2.11. Consider the connection form β j and curvature form Ω j on Λ+. Consider alsothe constants C±

i =C±i j (equivalently, the linear map C±=C±

j :Rk →R) involved in the definitionof the connection (Lemma 4.2.1). We write dM j for the volume form α1 ∧·· ·∧αk ∧dαn

j . Then,for appropriate choice of constants C±

i j we have∫M

βj(X j)dM j = 0

Before we prove this lemma, it will be necessary to give a new definition. This definitionwill be used to choose the appropriate constants C±

i j

Definition 4.2.12. [Entropy of a invariant subbundle] Let F be a sub-bundle of T M that isinvariant by a flow ψt . Choose a volume form dx of M and a never vanishing section ζ of thedeterminant bundle of F , that is of the volume forms of F (if necessary, quotient the bundlemodulo sign). We define, for t ∈ R

at = at(F) =∫

Mlog |det(dψt)F |dx

where the determinant is taken with the help of the section ζ . Now, observe that fort,s ∈ R, we have:

at+s =∫

Mlog |det(dψt+s)F |dx

=∫

Mlog |det(d(ψt ∘ψs))F |dx =

∫M

log |det(dψtdψs)F |dx

=∫

Mlog |det(dψt)F det(dψs)F |dx =

∫M

log |det(dψt)F |+ log |det(dψs)F |dx = at +as

This means that at is a continuous one parameter subgroup of R. Therefore, there existss= s(F) ∈ R such that

at = s ·Vol(M) · t

where Vol(M) =∫

M dx. Notice that s is does not depend on the choice of the section. The quantitys is called the entropy of the sub-bundle F .

3 More technically, as our bundle is one dimensional, then our connection form has values on a onedimensional Lie algebra, which is therefore, abelian, and thus, β ∧β (A,B) = [β (A),β (B)] = 0


Now, we consider the entropy s±j of the fiber bundles E± invariant by the flow φj

t of X j

(the volume form is dM j). As E+⊕E− admits a non degenerate bilinear antisymmetric formthat is invariant by φ (the form Θ), we obtain that the entropy of E+⊕E− is zero4 , that is

s+j + s−j = 0

We choose our constants C±i j such that

δi js±j

n=C±

i j (4.9)

Proof of Lemma 4.2.11. In what follows we avoid using the index j that indicates which con-nection we are using. But whenever the index j is used, it indicates the very same index of theconnection.

Let τ(t) be the parallel transport of ∇ along the orbits of the flow of X j j starting at thepoint p0. Let φt be the flow along X j, and let p = φt(p0). From the definition of ∇, a vector fieldY ∈ E+ is parallel with respect to φ if

0 = ∇φt

Y = ∇X jY = [X j,Y ]+C+j jY

If we denote

s =C+j j =

s+jn

Then Y is parallel with respect to φ if [X j,Y ]+ sY = 0. If we write Yp = τ(t)Y0 then,

τ(t)|E± = e−stdφt

In fact:

LX jY = LX j(e−stdφtY0)

=ddt(e−st)dφtY0 + e−stLX j(dφtY0)

But LX j(dφtY0) = 0 because (dφtY0)φs = dφt+sY0 and thus

lims→0

dφ−sdφt(Y0)φs(p0)−dφt(Y0)

s= lim

s→0

dφ−sdφt+s(Y0)−dφt(Y0)

s= lim

s→0

dφt(Y0)−dφt(Y0)

s= 0

4 Remember that a symplectic matrix S has determinant 1, and from

(dα j)p(u,v) = (dα j)φt(p)(dφtu,dφtv)

we obtain (det(φt)|E+⊕E−

)2= 1


On the other hand ddt (e

−st) =−se−st , and thus, [X j,Y ] =−sY and therefore Y is parallelas we wanted.

Now, let ∆ = ∆ j(t) be the determinant of the parallel transport restricted to E+ (thesection ζ is used to compute the determinant). We have

∆(t) = e−nstdet(dφυt |E+)

From ∇X j = 0 we obtain

β (X j)ζ (X j) = (∇X jζ )(X j) = X j(ζ (X j))

Also, as ζ was chosen to be never vanishing, we have:

βj(X j) =

X j(ζj(X j))

ζ j(X j)

=ddt

log(∆(t))

=ddt

(log(det(dφt)|E+)

)−ns

=ddt

(log(det(dφt)|E+)

)− s±

But, s+ is defined by

s+Vol(M)t =∫

Mlog(det(dφt)|E+)dM j

Taking the derivative with respect to t on both sides we obtain∫M

βj(X j)dM j = 0

Lemma 4.2.13. For every p ∈ 1, . . . ,n, we have (Ω j)p ∧dα jn−p = 0.

Proof. We only prove in the case j = k. The other cases are similar. It is clear that the formΩ = Ωk is invariant by the action, but by the topological transitivity of the action, there is aunique volume form invariant by it, and thus, there exists constants cp such that

α1 ∧·· ·∧αk ∧Ωp ∧dαk

n−p = cpdMk

If these constants cp are zero, the proof is finished. So we shall suppose that they are notzero and α1 ∧·· ·∧αk ∧Ωp ∧dαn−p is a true volume form on M.


Thus, using Ω = dβ and integrating by parts, we have:

cp

∫dMk =

∫α1 ∧·· ·∧αk ∧∧Ω

p ∧dαkn−p

=∫

β ∧α1 ∧·· ·∧αk−1 ∧Ωp−1 ∧dαk

n−p+1

Now, we write β = ∑ j f jα j +η where η vanishes on T φ . Clearly5

β ∧α1 ∧·· ·∧αk−1 ∧Ωp−1 ∧dαk

n−p+1 = fkα1 ∧·· ·∧αk ∧Ωp−1 ∧dαk

n−p+1

and from η|T φ= 0 it follows β (Xk) = fk. And thus

cp

∫dMk =

∫β

k(Xk)α1 ∧·· ·∧αkΩp−1 ∧dα

n−p+1

= cp−1

∫β (Xk)dMk = 0

as wanted.

We are finally ready to prove Theorem 4.2.10:

Proof of Theorem (4.2.10). First we notice that B j leaves E+ invariant (this follows from Lema4.2.5). Consider now the eigenvalues ξ1, . . . ,ξn of B j|E+ . It follows from the Lemma 4.2.13 that,for every p ∈ 1, . . . ,n we have

∑i1<...<ip

ξi1 . . .ξip = 0 (4.10)

And thus, ξi = 0 for every i, and thus B j is nilpotent.

4.2.3 The Lie algebra g′

We have already studied the Lie algebra of Killing fields of a general rigid geometricstructure in the sense of Gromov. Now, we devote this section to the study of the particularalgebra that arises from our particular geometric structure σ associated with a contact pair and5 We have seen that K(Xs, · ) = 0 for every s ∈ 1, . . . ,k, and thus, iXsΩ = 0. Thus if we write

β = ∑ j f jα j +η , then

β ∧α1 ∧·· ·∧αk−1∧Ωp−1 ∧dαk

n−p+1 =

= fkα1 ∧·· ·∧αk ∧Ωp−1 ∧dαk

n−p+1 +ηα1 ∧·· ·∧αk−1 ∧Ωp−1 ∧dαk

n−p+1

but ηα1 ∧ ·· · ∧αk−1 ∧Ωp−1 ∧ dαkn−p+1 is a also an action invariant volume form, and thus differs

from dMk by a constant. As iXk ηα1 ∧·· ·∧αk−1 ∧Ωp−1 ∧dαkn−p+1 = 0 it follows that this constant is

null.


an Anosov action. In the first section we remember and fix some notations, we also show thatthe Killing vector fields are infinitesimal affine transformations. In the second section we showthat the Lie algebra of Killing vector fields is reductive and in the third section, after some tech-nical preparation we give a explicit construction of the semisimple part of the Levi decomposition.

4.2.3.1 Compatibility and the Isotropy group

From Gromov’s open-dense orbit theorem (Theorem 2.2.17), we obtain an open dense setU , where the pseudo-group Autloc(σ) acts transitively. It is well known that compact orbits oftopologically transitive Anosov actions are dense. Moreover, compact orbits are homeomorphicto a torus. We shall take a point v0 ∈ U that has a compact orbit (do not confuse the orbit ofthe Anosov action with the orbit of local automorphism of the geometric structure). The tangentspace Tv0M will be denoted by V0.

As before, we take g′ the Lie algebra of germs of killing vector fields of σ at v0. Aspreviously observed, this is the Lie algebra of germs at v0 of vector fields Y such that

[Y,X j] = 0

LY dαi = 0

Notice that the first condition also implies [Y,E±]⊂ E±;

We take h′ ⊂ g′ the Lie subalgebra of (germs of) Killing vector fields that fixes v0. Aswe have seen, this is precisely the Lie algebra of the Lie group H ′ = Autloc

v0v0(σ).

We also take q′± to be the Lie subalgebras of the vector fields Y such that Yv0 ∈ (E±)v0 .And finally, we take p′± to be the Lie subalgebra q′±⊕RX1 ⊕·· ·⊕RXk.

Lemma 4.2.14. Every Killing vector field is an infinitesimal affine transformation for theconnections ∇ = ∇ j defined on Lemma 4.2.1.

Proof. From Lemma 2.3.13, we must check that every Y ∈ g′ satisfies

[LY ,∇Z] = ∇[Y,Z] for every vector field Z on M

that is, for every vector field W we have

0 = [LY ,∇Z]W −∇[Y,Z]W (4.11)

= [Y,∇ZW ]−∇Z[Y,W ]−∇[Y,Z]W (4.12)


As the above expressions are linear in W , we just have to prove that 4.11 is valid for W

tangent to each of the distributions T φ , E+ and E−. The linearity on Z give us the same freedom.

The case W = Xs for some s this is trivial, for ∇Xs = 0 and as Y ∈ g′, then [Y,Xs] = 0.

For W ∈ E± and Z = Xs for some s, we obtain

[Y,∇ZW ]−∇Z[Y,W ]−∇[Y,Z]W =

= [Y, [Xs,W ]+C±s W ]− ([Xs, [Y,W ]]+C±

s [Y,W ])

= [Y, [Xs,W ]]+ [Xs, [W,Y ]] = [W, [Xs,Y ]] = 0

Now, for W ∈ E± and Z ∈ E∓ we obtain

[Y,∇ZW ]−∇Z[Y,W ]−∇[Y,Z]W =

= [Y, p±[Z,W ]]− p±[Z, [Y,W ]]− p±[[Y,Z],W ]

= [Y, p±[Z,W ]]− p±([Z, [Y,W ]]+ [[Y,Z],W ])

= [Y, p±[Z,W ]]− p±([Y, [Z,W ]])

But, as [Y,E±]⊂ E±, we have, that for any vector field G, [Y, p±G] = p±[Y,G], and theabove equations equals to zero.

Finally, it remains to see the case Y,Z ∈ E±. For this we use the fact that dα = dα j isnon degenerated. We take G ∈ E∓ and we shall prove that

dα([Y,∇ZW ]−∇Z[Y,W ]−∇[Y,Z]W,G) = 0

First we remember, that for A ∈ E± and B ∈ E∓ we have dα(A,B) =−α([A,B]). More-over, for any vector field A = A++∑i aiXi +A− where A± ∈ E∓ we have

α j([Y,A]) = α j([Y,a jX j]) = Y (a j) = Y (α j(A)) (4.13)


Now we compute

dα([Y,∇ZW ],G) =−α([[Y,∇ZW ],G])

=−α([[Y,G],∇ZW ]+ [[G,∇ZW ],Y ]

)= dα([Y,G],∇ZW )−Y (dα(G,∇ZW ))

= Z(dα([Y,G],W ))−dα(∇Z[Y,G],W )

−Y (Z(dα(G,W )))+Y (dα(∇ZG,W ))

= Z(dα([Y,G],W ))−Y (Z(dα(G,W )))

−dα([Z, [Y,G]],W )+Y (dα([Z,G],W ))

= Y (Z(dα(W,G)))−Z(α([[Y,G],W ]))

+α([[Z, [Y,G]],W ])−Y (α([[Z,G],W ]))

We also compute

dα(∇Z[Y,W ],G) = Z(dα([Y,W ],G))−dα([Y,W ],∇ZG)

= α([[Y,W ], [Z,G]])−Z(α([[Y,W ],G]))

= α([Y, [W, [Z,G]]]+ [W, [[Z,G],Y ]])−Z(α([[Y,W ],G]))

= Y (α([W, [Z,G]])+α([W, [Z,G],Y ])−Z(α([[Y,W ],G]))

and using the Jacobi identity for 4 terms

[x, [y, [z,w]]]+ [y, [x, [w,z]]]+ [z, [w, [x,y]]]+ [w, [z, [y,x]]] = 0 (4.14)

we obtain

dα(∇[Y,Z]W,G) = [Y,Z](dα(W,G))−dα(W,∇[Y,Z]G)

= [Y,Z](dα(W,G))+α([W, [[Y,Z],G]])

= [Y,Z](dα(W,G))+α([G, [W, [Z,Y ]]]+ [Z, [Y, [G,W ]]]+ [Y, [Z, [W,G]]])

= [Y,Z](dα(W,G))+α([G, [W, [Z,Y ]]]+ [Z, [Y, [G,W ]]])+Y (α([Z, [W,G]]))

Summing every term we obtain

Z(Y (dα(W,G)))+Z(α([[Y,W ],G]+ [[G,Y ],W ])) (4.15)

+Y (α([[W,G],Z]))+α([G, [W, [Z,Y ]]]+ [Z, [Y, [G,W ]]]) (4.16)

+α([W, [Y, [Z,G]]])+α([W, [Z, [G,Y ]]]) (4.17)

Using the Jacobi identity and α j([Y,A]) = Y (α j(A)) the first line is zero and the last linesimplifies to α([W, [G, [Z,Y ]]]). Now we just have to use the 4 terms Jacobi identity (4.14) toconclude that

[G, [W, [Z,Y ]]]+ [Z, [Y, [G,W ]]]+ [W, [G, [Z,Y ]]] = [Y, [Z, [W,G]]]


Which reduces the above expression 4.15 to Y (α([[W,G],Z]))+α([Y, [Z, [W,G]]]). From 4.13,this is zero. Which finishes the proof.

It follows, from Lemma 2.3.15 that we can identify g′ with a subset of End(V0)⊕V0 viathe map

θo : g′ → End(V0)×V0

Y ↦→((AY )v0 ,Yv0

)where AY = LY −∇Y , and with the following Lie algebra structure:

[(A,a),(B,b)] = ([A,B]−R(a,b),Ab−Ba+T (a,b))

where T denotes the torsion tensor and R denotes the curvature tensor. The following lemmaallow us to better understand the map θo.

Lemma 4.2.15. Consider the map i : H ′ → GL(V0) given by i(φ) = dφv0 . Then, this map isinjective and di coincides with the restriction of θo to h′.

Proof. Injectiveness follows from rigidity. Now to prove the second assertion, we consider Y ∈ h′

and ψ t := exp(tY ). Let Z be a vector field around v0. As ψ t fixes v0, the parallel transport in thisorbit reduces to the identity, and therefore, (∇Y Z)v0 = 0. We have

die(Y ) =ddt

∣∣∣∣t=0

i(exp(tY )) =ddt

∣∣∣∣t=0

(ψ t*)

but, for a vector field Z around v0

limt→0

1t(ψ t

*Z −Z) = limt→0

1−t

(ψ−t* Z −Z) =−(LY Z)v0 = (∇Y Z)v0 − (LY Z)v0 = (AY )v0Zv0

and thus,die(Y ) = (AY )v0

as we wanted.

This lemma allow us to identify g′/h′ with V0 (and similarly, q′±/h′ with (E±)v0). Weshall also identify g′ and h′ with their images θo(g

′) and θo(h′). And also H ′ with j(H ′).

Moreover, notice that while the map θo depends on the connection (so it would be morecorrect to denote θ

jo ), the restriction of the map to h′ does not.

Let ϕt be the flow of X given by an Anosov element. As the Anosov elements form anopen cone on Rk, we have some freedom to choose this element. We shall chose an element,such that v0 is a periodic point of ϕt of period t0. Consider the map (dϕt0)v0 ∈ H ′ ⊂ GL(V0). AsH ′ is algebraic, it has a finite number of connected components, and thus, there exists n0 ∈ N


such that (dϕnt0)v0 is in the connected component (of H ′) of the identity. We set l0 = (dϕnt0)v0 ,and L0 the logarithm of the hyperbolic part of the Iwasawa decomposition of l06. In particularexpL0 have positive eigenvalues.

Lemma 4.2.16. The following assertions are valid:

1. H ′ is an algebraic subgroup of GL(V0)

2. L0 ∈ h′

3. We can choose an iterate lr0, r ≥ 0 of l0 such that the eigenvalues of the corresponding L0

(that is, the logarithm of the hyperbolic part of lr0) on E+

v0(resp. E−

v0) are strictly negative

(resp. positive).

Proof. 1. From the demonstration of the open-dense orbit theorem, it follows that

H ′ = Autlocv0v0

(σ) = Autrv0v0(σ)

for some r. As Autrv0v0(σ) is algebraic by construction, it coincides with its image on

GL(V0).

2. As H ′ is algebraic, it contains both the hyperbolic and the elliptic part of its elements(Helgason (HELGASON, 1979), IX.§7 Lemma 7.1, or Seco et al. (PATRãO; SANTOS;SECO, 2008).).

3. This will follows from the Anosov property of the action. In fact, as L0 ∈ h′, we usethe identification θo and write L0 = (L0,0), where L0 = (∇L0 −LL0)v0 . However, as wehave seen (Lemma 4.2.15), the restriction of θo to h′ coincides with the differential ofH ′ ∋ h ↦→ dhv0 ∈ GL(V0).

Thus, L0 acts on Z0 ∈ Ev0 precisely as the (hyperbolic part of the) differential of an Anosovelement. Let us call this differential T .

While it may not be true that‖T|E+v0

‖ ≤ 1

It is true that‖T r

|E+v0

‖ ≤ 1

for some r. Thus, T r restricted to E+v0

have eigenvalues of module less then one, andtherefore it’s hyperbolic part restricted to E+

v0have eigenvalues on the open interval ]0,1[,

and thus its logarithm have strictly negative eigenvalues.

A similar arguments works for the E−v0

case.

6 While it is not true, in general, that the logarithm of an element of a Lie group exist, the logarithm mapis well defined on the hyperbolic part of the Iwasawa decomposition


4.2.3.2 g′ is reductive

Theorem 4.2.17. The Lie algebra g′ is reductive and its centre is

I =⊕

j

RX j

This theorem will follow from the following lemma:

Lemma 4.2.18. The nil-radical of g′ is I .

Proof of Theorem 4.2.17. First observe that the lemma implies that I is in fact the centre of g′,just notice that the center is in fact a nilpotent ideal of g′. Consider a solvable ideal a of g′. Thenthere exists ideals a1, . . . ,al such that [a j,a j]⊂ a j+1 and

a= a0 ≥ a1 ≥ ·· · ≥ al ≥ al+1 = 0

In this setting, al is an abelian ideal, and therefore is nilpotent. From Lemma 4.2.18. Itfollows al ⊂ I .

Now, [al−1,al−1]⊂ al ⊂ I and thus

[[al−1,al−1],al−1]⊂ [I ,al−1] = 0

that is, al−1 is a nilpotent ideal of g′ and therefore al−1 ⊂I . Proceeding by induction, we obtaina⊂ I and thus I is in fact the radical of g′, which is therefore reductive.

Proof of Lemma 4.2.18. Notice that I is inside the centre Z(g′) by definition. Take a nilpotentideal j′. As expL0 is hyperbolic, this means that L0 has real eigenvalues and we can write

g′ =⊕i∈R

g′i ; j=⊕i∈R

ji ; Tv0M =V0 =⊕i∈R

V0,i

where g′i (respectively ji and V0,i) is the (generalized) eigenspace associated with the eigenvalue i

of the action of L0 on g′ (respectively j and V0).

Remember that we are identifying g′ with it’s image under the map θo : g′→End(V0)⊕V0.Also remember that h is identified with its image (via θo) on End(V0), and thus, we identifyL0 = (L0,0). Now, for i = 0 let (A,y) ∈ ji, then, for every (A′,y′) ∈ g′−i we have:

(A′′,y′′) =: [(A,y),(A′,y′)] ∈ j0

Notice that j0 ⊂ g′0, and

g′0 = (Q,q) ∈ g′; [L0,(Q,q)] = 0

= (Q,q) ∈ j;([L0,Q],L0q) = 0


but, the eigenvalues of L0 restricted to E+⊕E− are strictly non zero, and thus, g′0 ⊂h′⊕I . Now, as (A,y) ∈ ji for i = 0, we have that L0y = iy, and thus y ∈ E+⊕E−, it followsthat, α j(y) = 0. And analogous for y′, that is, (A′,y′) ∈ g′−i, and thus, L0y′ = −iy′ and thus,y′ ∈ E+⊕E−, therefore α j(y′) = 0

On the other hand, V0,0 ∩V+0 = 07 and, as (A,y) ∈ ji and (A′,y′) ∈ g′−i, this means, that

Ay′ = 0 = A′y. Moreover, either y = p+y and y′= p−y′ or y = p−y and y′= p+y′.

Therefore,

(A′′,y′′) = [(A,y),(A′,y′)] = ([A,A′]−R(y,y′),T (y,y′))

= ([A,A′]−R(y,y′),∑j

λ j([y′+,y−]+ [y′−,y+])X j +λ j(y)S j(y′)−λ j(y′)S j(y))

= ([A,A′]−R(y,y′),∑j

λ j([y′,y])X j)

Now, consider θo(X j) = ((AX j)v0,X j,v0). As

AX jY = LX jY −∇X jY

= LX jY − [X j,Y ]−S j(Y ) =−S j(Y )

we have, that θo(X j) = (−S j,v0 ,X j,v0) = (−S j,X j). Thus,

(A′′,y′′)−∑j

λ j([y′,y])θo(X j) = ∑j([A,A′]−R(y,y′)+λ j([y′,y])S j,0)

That is,

(A′′,y′′)Moda= ∑j([A,A′]−R(y,y′)+λ j([y′,y])S j,0)

Now, j is a nilpotent ideal, and therefore the adjoint of its bracket has null trace for everyinvariant subspace stable (Foulon, P. and Labourie, F. (FOULON; LABOURIE, 1989). Proof ofProposition 3.7) by j, that is

Tr|E±v0(A′′,y′′) = 0

Notice that I is contained in the centre, and therefore it doesn’t contribute to the trace,and thus

Tr|E+v0(A′′,y′′)ModI = 0

From the linearity of the trace, we have

0 = Tr|E+v0[A,A′]+Tr|E+

v0R(y,y′)+∑

jTr|E+

v0α j([y′,y])S j

7 We just observe that V+0 =

⊕i>0V0,i.


The first term is clearly zero, while the trace of a projection is given by the dimension of thesubspace it projects into, and thus TrE±

v0S j = nC±

j . We obtain

0 =dλ (B jy,y′)+∑i

αi([y′,y]nC+i

= dα j(B jy,y′)+n∑i

C±i dαi(nC+

i y,y′)

But, we have chosen the constants C±i =C±

i j =δi js

±j

n and thus

0 =dα j((B j + s+j )y,y′)

As this is true for every y′, it follows (B j + s+j )y. But s+j = 0 and B j is nilpotent (andthus doesn’t have a non zero eigenvalue), therefore y = 0.

We conclude that ji ⊂ h′8. Thus,

j⊂ h′⊕I

Now, h′ can’t contain an ideal of g′9, so h′ ∩ (j⊕I ) = 0 and we obtain j ⊂ I aswanted.

4.2.3.3 The semisimple part of the Levi decomposition

4.2.3.3.1 Flatness of the determinant Bundle

Before we construct the semisimple part of the Levi decomposition we must first returnto a technical aspect. We finish the proof of the Theorem 4.2.8

Definition 4.2.19. Let g be a Lie algebra, a Cartan subalgebra c of g is a nilpotent subalgebrathat is equal to its own normalizer, that is

[X ,Y ] ∈ c ∀ Xc =⇒ Y ∈ c

Lemma 4.2.20. There exists a Cartan subalgebra c′ of g′ such that c′ ⊂ h′⊕I

The proof will use the following classical Lie algebra theory, those results will be statedwithout proof.8 We proved that for (A,y) ∈ ji, y = 09 Let (A,0) ∈ h′, and chose q ∈V0 such that Aq = 0, we have

[(A,0),(0,q)] = (0,Aq) ∈ h′


Lemma 4.2.21 (Bourbaki, (BOURBAKI, 2005), VII.§2.3 Proposition 10). Let g be a Lie algebra,a an abelian Lie subalgebra and b the commutant of a in g. Then, the Cartan subalgebras of bare the Cartan subalgebras of g containing a.

(Proof of lemma (4.2.20)). We take a= SpanRL0, and b := y ∈ g; [y,L0] = 0= g′0. We takec a Cartan subalgebra of g′0. From the previous lemma, this is a Cartan subalgebra of g′ containingL0. As c⊂ g′0 ⊂ h′⊕I the proof is finished.

Lemma 4.2.22. The linear operator B j defined on 4.8 is semisimple.

Proof. As we have seen, there exists a Cartan subalgebra c′ of g′ contained in h′⊕I . FromLemma 3.3.25, c′C is a Cartan subalgebra of g′C, and from Lemma 3.3.35, (c′C,g

′C) is splitting.

Consider the corresponding root space decomposition:

g′C = c′C⊕⊕

λ∈∆(c′C,g′C)

(g′C)λ

As C is algebraically closed, (g′C)λ is one-dimensional.

Now we identify Tv0M =V0 = g′/h′, and consider the action of B j on V0, it commuteswith the action of H ′, for H ′ is the space of isometries that fixes v0. Therefore, it commutes withthe action of c′ (It commutes with the action of X j trivially, and thus with the action of h′⊕I ).

We can consider the adjoint action of c′C on g′C and, thus, on g′C/h′C. As we observed,

the root spaces on g′C/h′C are unidimensional. As the action of B j commutes the adjoint action

of c′C, they share the same invariant subspaces and, therefore, the eigenspaces of B j are alsounidimensional. Thus, B j is semisimple.

We are now ready to prove

Theorem 4.2.23 (Theorem 4.2.8). The bundle Λ± endowed with the connection ∇ j has zerocurvature.

Proof. We defined the operator B j by Ω j(Y,Z) = dα j(B jY,Z). From Theorem 4.2.10 B j isnilpotent, and from Lemma 4.2.22, it is also semisimple. Therefore B j = 0 and Λ± has curvaturezero.

Corollary 4.2.24. As the bundle Λ+ is flat, it must have a unique (up to a constant) parallelsection ζ j.


4.2.3.3.2 A semisimple Lie algebra

In the previous paragraph, we showed the existence of an unique (up to a constant)parallel sections ζ j of the determinant bundle Λ+.

Definition 4.2.25. We define the maps dχ j : g′ → R by:

∀Y ∈ g′ LY (ζ j) = dχ j(Y )ζ j

The uniqueness of ζ j means that this is a well defined map

.

Lemma 4.2.26. For X ∈ I and ζ a section of Λ+ we have

∇jX ζ −∇

iX ζ = (αi(X)s±i −α j(X)s±j )ζ

Proof. First, we observe that ∇jXY± = [X ,Y±]+α j(X)C±

j jY±, and thus,

∇jXY±−∇

iXY± = (α j(X)C±

j j −αi(X)C±ii )Y

±

Now we take Y1, . . . ,Yn local basis for E±, from the formula

∇X ζ (Y1, . . . ,Yn) = X(ζ (Y1, . . . ,Yn))−∑l

ζ (Y1, . . . ,∇XYl, . . . ,Yn)

we obtain

(∇jX ζ −∇

iX ζ )(Y1, . . . ,Yn) = ∑

lζ (Y1, . . . ,∇

iXYl −∇

jXYl, . . . ,Yn)

= ∑l(αi(X)C±

ii −α j(X)C±j j)ζ (Y1, . . . ,Yl, . . . ,Yn)

= (αi(X)s±i −α j(X)s±j )ζ (Y1, . . . ,Yl, . . . ,Yn)

Proposition 4.2.27. Let us write ζi = fi jζ j then X( fi j) = 0 if, and only if αii(X)si = α j(X)s j.

Proof. From the previous lemma,

(αi(X)s±i −α j(X)s±j )ζi = ∇jX ζi −∇

iX ζi

= ∇jX ζi = ∇

jX fi jζ j

= X( fi j)ζ j + fi j∇jX ζ j

=X( fi j)

fi jζi


that is X( fi j) = 0 if and only if

αii(X)si −α j(X)s j = 0

.

Remark 4.2.28. From the previous lemma, we obtained that, for i = j,

X i( fi j)

fi j= si

Lemma 4.2.29. For every 1 ≤ j ≤ k, let dχ j : g′ → R be the character10, given by Definition4.2.25. We have:

LX jζ j = s jζ j

Moreover dχ j(L0) = Tr|E± (L0)> 0.

Proof. First of all, notice that dχ j is in fact a Lie algebra morphism. In fact,

dχ j([Y,Z])ζ j = L[Y,Z]ζ j = [LY ,LZ]ζ j

= LY LZζ j −LZLY ζ j

= LY (dχ j(Z)ζ j)−LZ(dχ j(Y )ζ j)

= dχ j(Y )dχ j(Z)ζ j −dχ j(Z)dχ j(Y )ζ j = [dχ j(Y ),dχ j(Z)]ζ j

As wanted. Now, let’s prove dχ j(X j) = s j.

We shall writeY = (Y0, . . . ,Yn),

Y (k) = (Y0, . . . ,Yk−1,Yk+1, . . . ,Yn)

and for indexes α1 < .. . < αl , l < n

Y (α1···αl) = (Y0, . . . ,Yα1−1,Yα1+1, . . . ,Yαl−1,Yαl+1, . . . ,Yn)

Let ∇ be a connection on ΛnE± and ζ be a (closed) flat section of ΛnE±.

0 = ∇Y0ζ (Y (0)) = Y0(ζ (Y (0)))−∑k

ζ (Y1, . . . ,∇Y0Yk, . . . ,Yn)

0 = dζ (Y ) =n

∑k=0

(−1)kYk(ζ (Y (k)))+∑i<k

(−1)i+kζ ([Yi,Yk],Y ik)

=n

∑k=0

(−1)kYk(ζ (Y (k)))+ ∑0<i<k

(−1)i+kζ ([Yi,Yk],Y ik)+

n

∑k=1

(−1)kζ ([Y0,Yk],Y 0k)

10 Once again we avoid the usage of the superscript ”± ”. The more correct notation is dχ±j


Also

d(iY0ζ )(Y (0)) =n

∑k=1

(−1)k−1Yk(ζ (Y0,Y (0k)))+ ∑0<i<k

(−1)i+kζ (Y0, [Yi,Yk],Y (0ik))

= Y0(ζ (Y ))−n

∑k=0

(−1)kYk(ζ (Y (k)))− ∑0<i<k

(−1)i+kζ ([Yi,Yk],Y (ik))

= Y0(ζ (Y ))+n

∑k=1

(−1)kζ ([Y0,Yk],Y 0k)

=n

∑k=1

ζ (Y1, . . . ,∇Y0Yk − [Y0,Yk], . . . ,Yn)

Now, we notice that for Y0 =X j, ∇=∇ j and ζ = ζ j we have ∇jX jY = [X j,Y ]+C+

j j p+Y +C−

j j p−Y ,

thus, if we take Y1, . . . ,Yn a basis of E+ we obtain ∇jX j

Yl − [X j,Yl] =C+j jYl for every l = 1, . . . ,n,

that is

LX jζ (Y(0)) = d(iX jζ )(Y

(0)) =n

∑k=1

ζ (Y1, . . . ,C j j+Yk, . . . ,Yn) = nC+j jζ (Y

(0))

but nC+j j = s j(= s+j ) and thus LX jζ j = s jζ j as we wanted.

Now, for any Y ∈ h′ we have dχ(Y ) = Tr|E± (Y ). Here we understand Y as an elementon End(V0). As this immersion actually depends on the connection11, the trace may depend onthe choice of Anosov element. However, as we have seen, when restricted to h′ the map θ0 isactually de differential of the map j : H ′ → GL(V0), j(h) = dhv0 which does not depend on thechoice of the connection. Thus, for Y ∈ h′ we have

dχ(Y ) = Tr|E± (Y ) (4.18)

and the trace formula doesn’t depends on the choice of Anosov element and associated connection,etc.

Corollary 4.2.30. We have:

X i − siL0

dχi(L0)∈ kerdχi ∩ (kerdχ j)

c i = j

Proof. First, we notice that dχ1(L0) = dχ2(L0) and write ri =si

dχi(L0). We also write f = fi j

(that is: ζi = f ζ j).

That X i − riL0 ∈ kerdχi is clear. Now notice that

LX j−r jL0ζi = LX j f ζ j − r jdχi(L0)ζi

= X j( f )ζ j + f LX jζ j − r jdχi(L0)ζi

=(X j( f )

f+ s j −

s jdχi(L0)

dχ j(L0)

)ζi =

X j( f )f

ζi = 0.

11 Remember, the map θ0 : g′ → End(V0)×V0 is defined as θ0(Y ) := (LY −∇Y ,Yv0)


Corollary 4.2.31. For any i, j ∈ 1, . . . ,k, i = j we have dχ j(Xi) = 0.

Proof. From previous lemmas, we have

LX jζi = LX j( fi jζ j) = X j( fi j)ζ j + fi jLX jζ j

=

(X j( fi j)

fi j+ s j

)ζi

=

((αi(X j)si −α j(X j)s j

)λ

i(X j)+ s j

)ζi = (0− s j + s j)ζi = 0

Remark 4.2.32. Notice that the Corollary 4.2.30 implies that the hyperplanes kerdχ j, j =

1, . . . ,k, are in general position, that is ∩ j kerdχ j have co-dimension k. To see this, we noticethat for every j, (

X j −s jL0

dχ j(L0)

)∈ ∩ j kerdχ j

It is clear that X1 − s1L0dχ1(L0)

, . . . ,Xk − skL0dχk(L0)

are linearly independent.

Remark 4.2.33. Consider the Lie algebra ∩ j kerdχ j. Then, the map ∩ j kerdχ j → g′/I is anisomorphism. In fact, from the previous remark this map is indeed between spaces of the samedimension, thus, we just have to prove that it is an injection. Now, let X = ∑l blXl be an elementof I . Then

dχ j(X) = ∑l

bldχ j(Xl) = b js j

That is, X ∈ ∩ j kerdχ j ⇔ b j = 0∀ j, that is the map is injective.

Corollary 4.2.34. The Lie algebra g is semisimple

Proof. From the previous remark g= g′/I and thus is semisimple.

We can also build a Lie group H, by taking characters χ j : H ′ → (R, · ) defined by

h*ζ j = χ j(h)ζ j

and taking H = ∩ j ker χ j. It is clear that Lie(H) = g∩ h′ = h. It is clear that H is a closedsubgroup of H ′. Moreover, as H ′ is algebraic, then so is H.

4.2.4 Building the Model Space

Our goal is to build a model space of our manifold M, that is, we shall build a homo-geneous space V = G′/H ′ such that M is locally modelled on V by G′ maps and the (G′,V )

structure is complete. First let’s remember the necessary ingredients.


∙ We have the Killing Lie algebra of our geometric structure, that is, the germs at v0 (apoint with a compact orbit) of vector fields Y such that [Y,X j] = 0 and LY dα j = 0. Weremember that the first condition also implies, [Y,E±]⊂ E±, and together with the secondcondition they implies LY α j = 0. The connected simply connected associated Lie groupis G′.

∙ We have the subalgebra h′ ⊂ g′ of vector fields Y such that Y0 = 0. This is precisely de Liealgebra of the isotropy group of v0: H ′ = Aut loc(v0).

∙ We have the subalgebras q′± of vector fields Y ∈ g′ such that Yv0 ∈ E± and the Liesubalgebras p′±= q±⊕I

∙ We have the Lie algebra g which is isomorphic to g′/I , and a Lie group H such that

Lie(H) = h= g∩h′

∙ We have the associated connected Lie subgroups H ′, G, H, Q′± ⊂ G′

Remark 4.2.35. Notice that if G0 is the connected, simply connected, Lie group associated withg, then G0 ×Rk is a connected, simply connected, Lie group with Lie algebra g⊕Rk = g′, andthus, G′ = G0 ×Rk, and G0 = G, that is G is also simply connected.

Proposition 4.2.36. The subgroup H ′ is closed in G′

Before proving this proposition, let’s see a technical lemma

Lemma 4.2.37. Consider G′ = G×Rk a Lie group and H ′ ⊂ G′ be a Lie subgroup. Then, H ′ isclosed in G′ if, and only if, H ′∩G is closed in G.

Proof. Notice that we only need to prove the lemma for the case k = 1. In fact, if the lemmais true for k = 1, by induction, suppose that for 1 ≥ l we have H0 ⊂ G0 ×Rl is closed if andonly if H0 ∩G0 is closed. Then, let H ′ ⊂ G×Rl+1 such that H ′∩H is closed, then, by induction,H0 = H ′∩G×Rl is closed, and thus, applying the induction hypothesis again we obtain H ′ isclosed in (G×Rl)×R= G×Rl+1.

Now we prove the case k = 1.

Consider g′ = g⊕R the Lie algebra of G′, and h′ the Lie algebra of H ′. If h′ ⊂ g thenH ′ ⊂ G and there is nothing to prove. Now, suppose that h′ ⊂ g, then, there exists an elementv ∈ h with non zero R component. Take R = exp(Rv). The non zero R component of v impliesthat R is closed in H ′ and in G′. We have the inclusion H ′/R ⊂ G′/R, and the identifications

H ′/R = G∩H and G′/R = G


As the quotient map π : G′ → G′/R is continuous, the inverse image of closed sets areclosed, and thus, π−1(G∩H) = H ′ is closed.

Proof of Proposition 4.2.36. First, lets recall that H ′ is an algebraic subgroup of GL(Ev0). Wealso have that g′/h′ = Ev0 =V0 therefore the map

j : H ′ → Aut(g′)

h′ ↦→ (Y ↦→ h′*(Y )),

is injective 12.

That is, we can identify both H and H ′ with an algebraic subgroups of Aut(g′) andtherefore closed.

As g is semisimple, the map Ad : G → Aut(g′) is a covering onto it’s image. This impliesthat Lie(G) = Lie(Ad(G)) and therefore

Lie( j(H)) = Lie(H) = h= Lie(H) = Lie(Ad(H))

And thus (Ad(H))0 = j(H)0. As H is connected, that is H is closed in G. By our previouslemma H ′ is also closed in G′.

Remark 4.2.38. As H ′ is closed we can take the homogeneous space V = G′/H ′ we considerthe base point v0 ∈ V the induced flows on V given by

ϕj

t (v) = exp(tX j) · v

Clearly those flows commute and they define an action of Rk on V . We shall call it theAnosov action on V .

Remark 4.2.39. We consider the associated vector fields X j. We also consider E± the tangentdistribution to the foliation F± := gQ±/H ′|gQ± ∈ G′/Q±. We will call this tha strong stable(unstable) foliations on V .

It is clear that T (V ) = E+⊕RX1 ⊕·· ·⊕RXk ⊕ E−.

12 In fact, consider Y = (A,Y0) ∈ g′ ⊂ End(V0)×V0. Then, h′*Y = (dh′v0∘A,dh′v0

Y0). But g′/h′ = V0means that the the composite map g′ → End(V0)×V0 → V0 is surjective, and thus h′*Y = Y for allY ∈ g′ implies dh′v0

Y0 = Y0 for all Y0 ∈ V0 and thus dh′v0= Id. But we have already seen (Lemma

4.2.15) that the map H ′ ∋ h′ → dh′v0∈ GL(V0) is injective, and thus, h′ must be the identity.


Finally, for Y,Z ∈ g′ the alternating bilinear maps

B j(Y,Z) := (dα j)v0(Yv0,Zv0)

is well defined and H ′-invariant, which alow us to extends them to 2− forms ω j on V .

The map

θ(exp(Y ) · v0) = exp(Y ) · v0

is a local diffeomorphism between a neighbourhood of v0 and a neighbourhood of v0 such thatθ*X j = X j, θ*E± = E± and θ *ωi = dαi.

The transitivity of Autloc(σ) on Ω gives us local charts from Ω to G′/H ′ which preservessends (E±,X j,dα j) to (E±, X j,B j).

Remark 4.2.40. The local charts we defined above do not yet define a (G′,V )-structure on Ω,because we do not have any control of the transition maps, that is, we obtained an atlas

fi : Ui →Vi ⊂ V (4.19)

of Ω, and the change of coordinates of this atlas

fi ∘ f−1j : f j(Ui ∩U j)→ fi(Ui ∩U j)

preserves the (E±, X j,B j) -structure. However, we do not know if the transition maps fi ∘ f−1j

are the restriction of an element of G′.

In what follows, we shall construct a (slightly) larger group G′, such that the atlas 4.19will in fact be an atlas of a (G′,V )-structure on Ω.

Define the group G′ of diffeomorphisms of V that preserves E±, X j, and ω j for everyj. This is a Lie group and its identity connected component is precisely the image of G′ inDi f f (V ). Define H ′ the isotropy group of v0 in G′. The following lemma proves that there is noconfusion13.

Lemma 4.2.41. We have the following identifications

H ′ ≡ Autloc(v0)≡ A ∈ Aut(g′)|A(q±)⊂ q±;A(h′)⊂ h′;A(X j) = X j ω j ∘ (A×A) = ω j 13 Remember that we have previously defined H ′ as the isotropy group of our chosen point v0 ∈ M, that

is H ′ = Autlocv0(σ). As the map θ above is a local diffeomorphism from a neighbourhood of v0 to a

neighbourhood of v0 that preserves the structure, we identify Autloc(v0) = Autloc(v0)


Proof. First we observe that H ′ ⊂ Autloc(v0). This is clear, as global diffeomorphisms can berestricted to local ones.

Lets call the right side of the above equivalence H1. Remember that Autloc(v0) is thegroup of germs at v0 of local diffeomorphisms of V that preserves X j, E± and dα j, and thus, wecan identify

Autloc(v0) = Autlocv0(σ)

As we have already seen (proof of Proposition 4.2.36), the map

j : Autlocv0(σ)→ Aut(g′)

h′ ↦→ (Y ↦→ h′*(Y )),

is injective. Thus we can understand Autloc(v0) as a subgroup of Aut(g′). It is clear that theimage of j is contained in H1. Thus we have the following inclusions

H ′ ⊂ Autloc(v0)⊂ H1

Now, as G′ is simply connected,

Aut(g′) = Aut(G′)

and thus, an element A of H1 is also an automorphism of G′. The condition A(h′)⊂ h′ impliesthat it preserves H ′, and thus induces an diffeomorphism of V . The other conditions on A meansthat it preserves the A-structure, which shows that H1 ⊂ H ′.

Corollary 4.2.42. Every diffeomorphism from a connected open set of V to another that pre-serves X j, E± and ω is a restriction of an element of G′

Proof. It suffices to prove in the case of a diffeomorphism that fixes v0. The corollary followsfrom the equality H ′ = Autloc(v0).

Remark 4.2.43. Notice that we can identify V = G′/H ′ = G′/H ′. As H ′ is algebraic, it followsthat H ′ and G′ have a finite number of connected components. This means that up to finitecovering of M, we can suppose that G′ = G′.

Lemma 4.2.44. The diffeomorphisms θ from an open set of Ω to an open set of V such thatθ*X j = X j, θ*E± = E± and θ *ω j = dα j forms a maximal atlas of a (G′,V )- structure over Ω

Proof. By construction the domains of definition of those diffeomorphisms covers Ω, and fromthe previous lemma, it follows that the change of local coordinates are given by the elements ofG′.


Corollary 4.2.45. For every simple connected open set O ⊂ Ω there exists an "developing map"θ : O → V . That is, a local diffeomorphism θ such that θ*X j = X j, θ*E± = E± and θ *ω = dλ .

Proof. classical

Remark 4.2.46. As we have already observed in the chapter introduction, this developing mapis compatible with the Anosov action. It is clear that the induced action on the local model G′/H ′

is given by

Rk ×G′/H ′ → G′/H ′

(t1, . . . , tk,gH ′) ↦→ gH ′ exp(t1X1 + · · ·+ tkXk)

In particular, should Ω be equal to M, and the developing map M → G′/H ′ be a diffeo-morphism, then, the canonical representation ρ : π1(M)→ G′ give a diffeomorphism

M = π1(M)∖M → ρ(π1(M))∖G′/H ′

which smoothly conjugates our original action with an quasi-algebraic one.

We can build a connection ∇ on V in a similar to the one on M. It is clear that G′ is agroup of affine transformation with respect to ∇. This connection is also compatible with theA-structures on M and V , which allow us to prove the following lemma:

Lemma 4.2.47. The geodesics of ∇ tangents to the distributions E+ are complete

Proof. First we recall, from Lemma 4.1.8, that, for a given connected cone of Anosov elementson Rk, the set

∆ = p ∈ M|F+(p)⊂ Ω and F−(p)⊂ Ω

where F±(p) = q ∈ M| limt→∓∞ d(ϕt(q),ϕt(p)) = 014, is dense on M.

Consider Y ∈ E+ with base-point p ∈ V . Lets prove that a geodesic with initial conditionY , integrates to up to time greater then 1. There exists a connected, simple connected open setO ⊂ Ω, and a developing map θ : O → V and Y ∈ E+

v such that, θ*Y = Y . From the density of∆ and the transitivity of the pseudo-group Aut loc(σ), we can move the base point v of Y a littlebit and suppose that v ∈ ∆. Now, we have already proved that the geodesics in Ω tangents toE+ are complete, thus, they integrate up to a time greater the one. In particular, the geodesict → γ(t), starting from v with direction Y integrate up to time greater the one. As ∇E+ ⊂ E+,we have γ ∈ E+, thus, γ never leaves the leaf F+(v). As v ∈ ∆, γ(t) ∈ Ω, and we can supposethat γ([0,1])⊂ O15. The curve t ↦→ θ ∘ γ is our desired geodesic.14 Notice that F±(p) only depends on the choice of open Anosov cone.15 The developing map is defined for any simple connected open set, We just take O′ to be a tubular

neighbourhood of γ , and θ ′ the developing map of O′ that coincides with θ on the point v

4.3. Extending the Structure 123

Remark 4.2.48. At first we may be tempted to just use the completeness of the geodesics inM. But we build our (G′,V )-structure on Ω, which means that the developing maps are definedon the universal cover Ω, and not on M. That is, maybe our geodesic passes through a point inM∖Ω, and in this case, we won’t be able to develop our geodesic on V .

Our next section will solve this problem by extending the (G′,V )−structure to all M.

4.3 Extending the StructureFirst, we shall define appropriated local coordinates for M using the completeness of the

connection along the strong leaves.

Let ϕj

t , j = 1, . . . ,k be the flows of the vector fields X j. Define, for every v ∈ M the map

Ψv : TvM = E+v ⊕

⊕j

RX j ⊕E−v → M

Y = Y++ t1X1 + · · ·+ tkXk +Y− ↦→ Ψv(Y ) = ϕ1t1 ∘ · · · ∘ϕ

ktk(exp∇(τY+(Y−)))

where exp∇ is the exponential with respect to the connection ∇ and τY+(Y−) is the paralleltransport for time t = 1 along the geodesic t ↦→ exp∇(tY+) of the vector Y−.

We define, in an analogous way, for every v ∈ M and v ∈ V the maps Ψ:TvM → M andΨ:TvV → V .

From the previous section we see that those maps can be defined even for arbitrary largeY+,Y+ and Y+.

Definition 4.3.1. An open set O ⊂ M will be called A-star-shaped with respect to v ∈ O if thereexists an open set U ⊂ TvM such that

∙ Ψv is an diffeomorphism from U to O.

∙ If Y = Y++ t1X1 + · · ·+ tkXk +Y− is in U , then, for every s ∈ [0,1], so are

– Y++ t1X1 + · · ·+ tkXk + sY−

– sY++ t1X1 + · · ·+ tkXk

– t1X1 + · · ·+ st jX j + · · ·+ tkXk for every j ∈ 1, . . . ,k

In particular, O is contractible and therefore simple connected.

We have defined the A-star-shaped open sets in such way that they are well behaved withrespect to the affine transformations, that is, if O is an A-star-shaped open set with respect to v

and θ is a developing map from O to V , then

θ ∘Ψv = Ψθ(v) ∘Tvθ (4.20)


Lemma 4.3.2. Let v ∈ ∆ and O be an A-star-shaped open set with respect to v. Then, there existsan open dense subset O‘ ⊂ O∩Ω which is also A-star-shaped with respect to v.

Proof. We consider the open set U ⊂ TvM associated with the A-star-shaped set O and define

U ′ := Y = Y++ t1X1+ · · ·+ tkXk +Y− ∈U |

Ψv(Y++ t1X1 + · · ·+ tkXk + sY−) ∈ Ω ∀s ∈ [0,1]

We take O′ = Ψv(U ′). It is clearly an open set. Let us see that it is also dense.

For any open set W ⊂ Ω∩O take p ∈W ∩∆. We write p = Ψv(Y+0 + t01X1+ · · ·+ t0kXk+

Y−0 ).We must show that Ψv(Y+

0 + t01X1 + · · ·+ t0kXk + sY−0 ) ∈ Ω for every s ∈ [0,1].

Remember that, by definition, Ψv is defined by travelling a certain distance along theunstable foliation, then travelling a certain distance along the stable foliation, and acting a certainvector on the result.

As ∆ is invariant by the action, we have that Ψv(Y+0 +Y−

0 ) ∈ ∆.

By definition, the stable foliation of Ψv(Y+0 +Y−

0 ) is contained in Ω, in particular,Ψv(Y+

0 + sY−0 ) ∈ Ω. But Ω is invariant by the action, and thus

Ψv(Y+0 + t01X1 + · · ·+ t0kXk + sY−

0 ) ∈ Ω

It remains to show that O′ is A-star-shaped.

As Ω is invariant by the action and v ∈ ∆ ⊂ Ω, we have

t1X1 + · · ·+ tkXk ∈U ′ ⇔ Ψv(+t1X1 + · · ·+ st jX j + · · ·+ tkXk) ∈ Ω

that ist1X1 + · · ·+ tkXk ∈U ′ ⇔+t1X1 + · · ·+ st jX j + · · ·+ tkXk ∈U ′

We must now show, that for any s ∈ [0,1] we have

Y++ t1X1 + · · ·+ tkXk ∈U ′ ⇔ sY++ t1X1 + · · ·+ tkXk ∈U ′

that isY++ t1X1 + · · ·+ tkXk ∈U ′ ⇔ Ψv

(sY++ t1X1 + · · ·+ tkXk

)∈ Ω

But v ∈ ∆ and thus, it’s unstable foliation is in Ω. In particular, Ψv(sY+) ∈ Ω for every s.As Ω is invariant by the action, Ψv (sY++ t1X1 + · · ·+ tkXk) ∈ Ω

4.3. Extending the Structure 125

Lemma 4.3.3. Fix a background metric g on M and let us denote, for v ∈ M,r ∈ R>0 andY = Y++ t1X1 + · · ·+ tkXk +Y− ∈ TvM:

Br(v) = Y ∈ TvM ; ‖Y‖2 de f= g(Y+,Y+)+ |t1|2 + · · ·+ |tk|2 +g(Y−,Y−)< r2

and

Br(v) = p ∈ M; distg(c, p)< r

Then, there exists δ ,ε > 0 such that for every v ∈ M,

Ψv : Bδ (v)→ M

is a diffeomorphism onto it’s image and

Bε(v)⊂ Ψv(Bδ (v))

Proof. The proof is a variation of the usual argument for the exponential map and relies on thecompactness of M.

For each point p∈M consider εp < 1 such that (Ψp)|Bp(εp))is injective16. As injectiveness

is an open condition, there exists a neighbourhood p ∈Up ⊂ M such that

(Ψq)|Bq(εp))is injective

As M is a manifold, it is always possible to take a locally finite subcover Upii∈I . Wedefine h : M → R by

h(p) = minεpi; p ∈Upi.

This function is a locally constant positive function on M. Take g ∈C(M,R>0) such that0 < g(p)≤ h(p) for all p ∈ M.16 We must check that Ψv is in fact a local diffeomorphism at the origin. If ‖Y‖ is sufficiently small, then

the exponential map is a diffeomorphism onto it’s image and it’s differential at the origin is the identity,and we have, for Y = Y++ t1X1 + · · ·+ tkXk +Y−:

(dΨv)0Y =dds

∣∣s=0

(Ψv(sY )

)=

dds

∣∣s=0

(ϕ

1st1 ∘ · · · ∘ϕ

kstk(exp∇(τsY+(sY−)))

)= t1X1 + · · ·+ tkXk + ˜t1 · · · tk

dds

∣∣s=0

(τsY+(sY−)

)where ˜t1 · · · tk = Πt j =0t j. Now, γ(s) := τsY+(sY−) is a curve on T M which we write as γ(s) =(α(s),β (s)) where α(s) ∈ M and β (s) ∈ Tα(s)M. By definition, α(s) is actually the geodesic onM with starting vector Y+, and β (s) is the parallel transport of sY− to the desired point.It is easy tosee that d

ds

∣∣s=0

(τsY+(sY−)

)= Y++Y−


Define for any g ∈C(M,R>0) the submanifold

Ag = (p,Y ) ∈ T M ; ‖Y‖p < g(p)

and defineC := g ∈C(M,R>0) ; Ψ|Ag is injective

Where Ψ : T M → M×M is defined as

Ψ(p,Y ) = (p,Ψp(p,Y ))

We consider the map f : M → (0,1] defined as

f (p) = max1,r ∈ R ; Ψp|Bp(r) is injective

As C is not empty, we can also write

f (p) = maxg∈C

g(p)

As C is a family of continuous positive limited function, then so is f .

As M is compact, there exists 0 < δ ≤ 1 such that

f (p)≥ δ ∀p ∈ M

Now, we consider the map

M → R

p ↦→ maxr;Bp(r)⊂ Ψp(Bp(δ ))

By similar arguments this map is also continuous and positive and from the compacity ofM it admits a non zero minimum ε .

Corollary 4.3.4. For any dense subset U ⊂ M, we can cover M by A-star shaped open sets withbase points in U .

Proposition 4.3.5. Ω = M

Proof. We just need to prove that the (G′,V )-structure on Ω can be extended to M.

Using the Corolary 4.3.4, we take a cover Ovi of M by A-star shaped open sets withbasepoints vi ∈ ∆, and using Lemma 4.3.2we consider open, subsets O′

vi⊂ Ω∩Ovi(ε) dense in

Ovi and also A-star-shaped with respect to vi.

In particular, O′vi

is contractible, and therefore, simple connected. there exists, thus, adeveloping map θ : O′

vi→ V . We define θ : Ovi → V by

θ = ˆΨθ(vi) ∘ (Tviθ)∘Ψ−1vi

4.4. Completeness of the structure 127

It follows from 4.20 that θ is an extension of θ . From the density of O′vi

in Ovi , it followsthat θ*(X j) = X j, θ*(E±) = E± and θ *(ω) = dλ . We conclude that θ is a developing map17.That is, we managed to extend the (G′,V )−structure to M.

4.4 Completeness of the structureFinally we shall prove that the extended structure build in the previous section is complete,

that is

Proposition 4.4.1. Let M be the universal cover of M and let θ : M → V be a developing mapof the (G′,V )-structure. Then, θ is a covering map..

Proof. To prove our proposition, we must find, for each w ∈ V a neighbourhood O such thatθ−1(O) is the disjoint union of open sets Oii∈I such that the restriction of θ to each Oi is adiffeomorphism onto O.

Let w ∈ V and vii∈I = θ−1(w). Consider O an A-star-shaped open set with respect to w

and U ⊂ TwV the associated open set.

We define, for i ∈ I

Oi = Ψvi

((Tviθ)

−1(U))

It follows from 4.20 that θ induces an diffeomorphism θi from Oi to O.

Lets show that Oi ∩O j = /0 if i = j. For this, we consider the set

O′ = w′ ∈ O |θ−1i (w′) = θ

−1j (w′)

The set O′ is obviously closed18.

On the other hand, O′ is also open, for θi and θ j are the restriction of a θ to a certainopen sets. If there is a intersection (which is the case if θ

−1i (w′) = θ

−1j (w′) for some w′), then

θi = θ j on this intersection V ′, and thus θ(V ′)⊂ O′.

As w ∈ O′ if i = j and O is connected (actually contractible), it follows that O′ = /0.

Finally, we shall show that θ−1(O) = ∪i∈IOi

First we define the map

Φ : T M → T MY = Y++ t1X1 + · · ·+ tkXk + Y− ↦→ (ϕ1t1)* . . .(ϕ

ktk)*(τ(τY+Y−)(τY+Y )

)and in a similar way the map Φ : TV → TV . It is clear that17 A developing map is a local diffeomorphism (θ is the composite of local diffeomorphisms) that

preserves the invariant bundles and the 2-form.18 O′ is the intersection of two closed sets, the inverse images by continuous functions of a point


T θ ∘ Φ = Φ∘T θ

Moreover, Φ is a diffeomorphism19

Now, consider v′ ∈ V such that θ(v′) ∈ O. We can write θ(v′) = Ψw(Y ) for Y ∈ U .Consider

Y := (Φ)−1((Tv′θ)−1(Φ(Y ))

)and let v be the basepoint of Y . We have

T θ(Y ) = Φ−1 ∘T θ ∘ Φ(Y )

= Φ−1 ∘T θ(Tv′θ)

−1(Φ(Y )) = Φ−1(Φ(Y )) = Y

Thus the base point v must be vi for some i ∈ I. It remains to prove that

Ψvi(Y ) = v′

This will conclude the proof, for we have shown that v′ ∈ Oi.

Lemma 4.4.2. If Z has basepoint a, and Φ−1(Z) has basepoint b, then, Ψb ∘ Φ(Z) = a.

Proof. Notice that if we take an element TbM ∋ Y = Y++ t1X1 + · · ·+ tkXk + Y−, then, we canunderstand the map Ψb in the following way.

We take the geodesic with starting vector Y+ and transport along it, to time one, thevector Y−. Now we take a geodesic with this starting vector and take it’s time one and applyϕ1

t1 ∘ · · · ∘ϕktk .

19 in fact, we shall write explicit formulas for its inverse. Consider the smooth maps f±, g± : T M → T Mdefined by

f±(Y ) = τY±Y

g±(Y ) = τ−Y±Y

It is clear that f± ∘ g± = g± ∘ f± = Id. which makes f± diffeomorphisms. We can write

Φ(Y ) = (ϕ1t1)* . . .(ϕ

ktk)*( f− ∘ f+(Y ))

. Let Z = Z++ t1X1 + · · ·+ tkXk + Z− and let

(ϕ1−t1)* . . .(ϕ

k−tk)*(Z) := W

The inverse Φ−1 is given byΦ

−1(Z) = g+ ∘g−(W )

The key element here is the fact that if we write

Φ(Y++ t1X1 + · · ·+ tkXk + Y−) = Z++ s1X1 + · · ·+ skXk + Z−

, then s j = t j for every j.

4.4. Completeness of the structure 129

We can understand the map Φ in the following way. We take the geodesic with startingvector Y+ and transport along it, to time one, both the vector Y−. We now take a geodesic withthis starting vector and take it’s time one. We transport the original vector Y along both of thissegments, and apply D(ϕ1

t1 ∘ · · · ∘ϕktk) o the result. It is clear that Φ(Y ) will have basepoint Ψ(Y ).

We take Y = Φ−1(Z) and the result follows.

Remark 4.4.3. It is a classical theorem (Mostow, (MOSTOW, 1950), page 617, Corollary 1) inthe theory of homogeneous spaces that if G is a simply connected Lie group and H is a connectedclosed subgroup of G, then G/H is simply connected. Thus, the developing map of Theorem4.4.1 is in fact a diffeomorphism.

131

CHAPTER

5GROUP REDUCTION

The goal of this chapter is to study the quasi-algebraic action obtained in the last chapter.In particular, under some additional hypothesis, we obtain a reduction of the group G′, to asemisimple one of rank greater or equal to k. To be precise, our goal is to prove the followingtheorem

Theorem 5.0.1. Let (M2n+k,α,φ) be a generalized k-contact Anosov action. Suppose, moreover,that the action is faithful and H1(π1(M),R) = 0. Then it is smoothly conjugated to a quasialgebraic action (G,K,Γ,a) where G is semisimple of rank greater or equal to k and a iscontained in the Cartan subspace of Lie(G)

Step 1: Construct a semisimple lie group that acts transitively on M (Lemma 5.1.3)

Step 2: Show that the distributions tangent to the weak stable and weak unstable foliationscorresponds to opposite parabolic subalgebras p± of g (Theorem 5.1.9).

Step 3: Use the Langlands decomposition of the parabolic subalgebras p± to obtain a decompo-sition

g= h⊕L⊕n+⊕n−

where h is the centralizer of L. This requires some technical preparation and is done in

Step 4: Show that the composition map π1(M)→ G×Rk → G defines a discrete representationof π1(M). This is in fact the only step of the proof which uses the hypothesis that theAnosov action is faithful and H1(π1(M),R) = 0.

It is an easy consequence from the previous chapter that M = G′/H ′ = (G×Rk)/H ′,where G is semisimple.

In the Section 5.1 we show that subgroup G acts transitively on G′/H ′, which allows usto identify M = G/H, H = G∩H ′. We will also show that p± = g∩p

′± are opposite parabolic

132 Chapter 5. Group Reduction

subalgebras (Theorem 5.1.9) and their intersection correspond precisely to the algebra of thesubgroup which stabilizes an orbit (Corollary 5.1.7).

In the Section 5.2 we do some technical preparation, which we will use in Section 5.3 toobtain a decomposition

g= h⊕L⊕n+⊕n−

where h is the centralizer of L. And finally how that the composition map π1(M)→ G×Rk → G

defines a discrete representation of π1(M).

5.1 Parabolic subalgebrasLemma 5.1.1. The Lie algebra

g=k⋂

j=1

kerdχ j

has rank greater or equal to k

Proof. The goal of the proof is to construct an abelian subalgebra of g of hyperbolic elementsand of dimension k.

From Corollary 4.2.34 g= g′/I and thus is semisimple. To compute the rank of g, weobserve that

A0 :=L0

dχ j(L0)+

X1

s1+ · · ·+ Xk

sk∈ g

Now, we go back to the construction of L0, we considered a direction v of the Anosovaction such that v is an Anosov element and it’s associated flow has periodic orbit through thepoint v0. Let’s denote by Xv the associated vector field. We considered the differential of it’stime one map, some finite iteration of this map belongs to the group H ′, it is by constructionhyperbolic, and therefore we can take it’s logarithm to obtain the element L0 on the Lie algebrah′. By construction, the adjoint action of the element L0 is hyperbolic, and therefore so is theaction of A0 as defined above.

Notice that this construction uses the injection j : H ′ → Aut(Tv0M), j(h) = Tv0h, toidentify the group H ′ with it’s image. Without this identification, we do not need to take thedifferential of the time one map. We just take it’s logarithm. This means that we consider L0

not as an endomorphism of Tv0M but as a vector field on M. Clearly, exp(L0) = exp(Xv) andtherefore, Baker–Campbell–Hausdorff implies that L0 = Xv + Zv for some Zv ∈ g′ such thatexp(Zv) = Id.

If we consider a base of directions v,v2, . . . ,vk, such that each v j is an Anosov elementand the associated flows has periodic orbit through v0 (we remember that v0 has compact orbit,that is, it’s orbit is a k-dimensional torus, and thus such base of directions exists). Let us showthat L0,L2, . . . ,Lk are linearly independent. In fact,each L j is of the form Xv j +Zv j as the vector

5.1. Parabolic subalgebras 133

fields Xv j are linearly independent, ir remains to show that the space generated by the Zv j doesnot intercept the space generated by the Xv j . Let Z ⊂ g denote the vector space generated by theZV j and suppose that Z ∩I = 0, then there exists some linear combination L of L j such thatL ∈ I ∖0 which contradicts the fact that L ∈ h′ and therefore Lv0 = 0.

Now, let us show that the [Li,L j] = 0 for any i, j. As the action is abelian, the time onemaps commute with each other, and as they are hyperbolic, they are diagonalizable, thus, thereexists a basis of Tv0(M) such that the time one maps are simultaneously diagonalizable. Thelogarithm map is just the logarithm of each diagonal element, and therefore, the elements L j aresimultaneously diagonal. In particular, they commute with each other1.

Thus we can build elements

A j :=L j

dχ j(L j)+

X1

s1+ · · ·+ Xk

sk; 2 ≤ j ≤ k

such that, for i, j = 0,2,3, . . . ,k we have A j ∈ g, [Ai,A j] = 0 and by construction, each A j ishyperbolic. Therefore, the Cartan subspace of g is at least k dimensional, and g has rank k.

Lemma 5.1.2. The group G′ decomposes as a direct sum G′ = G×Rk where G is semisimpleof rank greater or equal then k and has Lie algebra g.

Proof. We define the group G by taking characters χ j : G′ → (R, · ) defined by

g*ζ j = χ j(g)ζ j

and taking G = ∩ j ker χ j. It is clear that Lie(G) = g.

From Lemma 5.1.1 g is semisimple with rank greater or equal than k.

If we denote by ϕ j the flow corresponding to X j then χ j(ϕit ) = δi jesi·t . Thus, the map

f : G×Rk → G′

(g, t1, . . . , tk) ↦→ g · ϕ1t1 · · · · · ϕ

ktk

is an isomorphism.

Lemma 5.1.3. The group G acts transitively on G′/H ′. In particular, this means we can identifyG′/H ′ = G/H.1 We note that we are identifying h′ with it’s image by the map θo : g′ → End(V0)×V0. We remember

that the Lie bracket on this image is given by

[(A,a),(B,b)] = ([A,B]−R(a,b),Ab−Ba−T (a,b))

and thus, the restriction of the bracket to θo(h′)⊂ End(V0)×0 is actually the usual commutator of

matrices (if we fix a given basis for V0).


Proof. We only need to prove that g′ = g+h′.

Consider the projection map

π : g′ → g′/g= (g⊕I )/g= I

. We will show that π restricted to h′ is surjective.

In fact, let h′ = π−1(π(h′)). Suppose that h′ $ g′, as g⊂ h′, this means that h′∩I hasco-dimension at least 1. But I corresponds to the Anosov action, and the base point we choose2

has compact orbit, therefore, there exists X ∈ I ∖h′ such that exp t0X ∈ H ′ for some t0. However,this contradicts the fact that tX ∈ h′ for t = 0.

Remark 5.1.4. In the previous proof we identified two different notions of exponential maps:The exponential map from a Lie algebra to its associated Lie group and the exponential mapthat associates a vector field on a manifold M to the flow it generates. This identification isnot possible in general, even when the Lie group corresponds to a group of diffeomorphismsof the manifold M. However, we are considering an homogeneous manifold G′/H ′, and thecorresponding Lie algebras, and the vector field we considered integrates to a one parametersubgroup of G′. In this context it is easy to see that both notions of exponential coincide.

Remark 5.1.5. We have previously define on G′/H ′ distributions E± tangent to the foliationsF± which corresponds to the strong stable and unstable foliations. We denote by I = RX1 ⊕·· ·⊕RXk.

Lemma 5.1.6. Consider the map π : G → G/H, and the distributions P± on T G given by

dπ(P±)⊂ E±⊕ I

Then, P± is the distribution obtained by left translation of the Lie subalgebra p± = p′±∩g.

Proof. In fact, let Q′± ⊂ G′ be the connected sub group associated with the Lie subalgebra q

′±

the strong foliations F± were constructed by left translating Q′±/H ′, and thus, the distribution

E± is precisely the left translation of q′±.

If we consider the connected subgroup P′± ⊂ G′ associated with the Lie subalgebra

p′±, we obtain that left translation of q

′± corresponds to the distribution tangent to the weakfoliations.

Corollary 5.1.7. The subalgebra p+ (resp p−) corresponds to the Lie subgroup of G thatstabilizes the weak stable (resp. unstable) leaf of eH ∈ G/H. In particular, the Lie algebra

g0 = (h′⊕I )∩g= p+∩p−

2 Remember that g′ is the Lie algebra of germs of local killing vector fields at fixed base point


is the Lie algebra of the stabilizer of the orbit (by the Anosov action on V (Remark 4.2.38)) ofeH.

Lemma 5.1.8. There exists a Cartan subalgebra c of g such that c⊂ g0.

Proof. It is known that if f : g1 → g2 is a surjective morphism of Lie algebras, and c1 is aCartan subalgebra of g1, then f (c1) is a Cartan subalgebra of g2. We consider the morphismπ : g′ → g′/I = g. It is clearly surjective, and from Lemma 4.2.20, there exists a Cartansubalgebra c′ ⊂ h′⊕I . It is clear that π(h′⊕I ) = g0, and thus, c= π(c′) is a desired Cartansubalgebra.

Theorem 5.1.9. Consider λ ∈ c*C and E ⊂ gC a vector subspace stable under the adjoint actionof cC. We denote

Eλ = Y ∈ E ; [Z,Y ] = λ (Z)Y ∀Z ∈ cC

Consider ∆(cC,E) = λ ∈ c*C∖0 ; Eλ = 0 Then, ∆(cC,p+) =−∆(cC,p

−).

Proof. Consider ∆0 = ∆(cC,(g0)C) and ∆± = ∆(cC,p±). As g= p++p−, it follows that

∆(cC,gC) = ∆+∪∆0 ∪∆−

where the union is disjoint.

Now, consider the complexification C⊗T M of the tangent bundle. The sections of thisbundle are called complex vector fields of M, and can be seen as derivations of complex functionson M, that is, we can identify

C⊗TpM = Derp(C∞(M,C))

Moreover, the space of complex vector fields is a Lie algebra, explicitly, it is the complexificationof the Lie algebra of real vector fields.

We also consider the complexified bundles Λr(C⊗T *M) = C⊗ΛrT *M. The exteriorderivative map

Γ(M,ΛrT *M)d→ Γ(M,Λr+1T *M)

induces a mapΓ(M,C⊗Λ

rT *M)dC→ Γ(M,C⊗Λ

r+1T *M)

which satisfies

dCω(Z0,Z1, . . . ,Zr) = ∑j(−1) jZ j(ω(Z0, . . . , Z j, . . . ,Zr))

−∑i< j

(−1)i+ jω([Zi,Z j],Z0, . . . , Zi, . . . , Z j, . . . ,Zr)


For a real vector bundle E over M the inclusion map E → C⊗E induces an inclusionmap Γ(M,E)→ Γ(M,C⊗E), the image via this map will be called a complexified section. Inparticular, the 1-forms α j can be complexified (we denote it by αC

j ) and it satisfies

dαCj (A,B) = A(αC

j (B))−B(αCj (A))−α

Cj ([A,B])

for any complex vector fields A,B.

Moreover, as E+⊕E− ⊂ ker(α j) then

C⊗ (E+⊕E−) = (C⊗E+)⊕ (C⊗E−)⊂ ker(αCj ) (5.1)

We have the identification p±C/g0C = E±v0

, which induces a identification

p±/g0 =(p±/g0

)C = E±

v0,C

For a fixed j, we have a non degenerate pairing dα j : E+v0⊕E−

v0→ R, which induces a

non degenerate pairingdα

Cj : C⊗E+

v0)⊕ (C⊗E−

v0)→ C

Moreover, as E±v0

are Lagrangian subspaces of dα j, then E±v0,C are also Lagrangian sub-

spaces of dαCj .

From Remark 3.3.37, gλ are at most one dimensional, and thus, there exists λ1, . . . ,λn ∈∆+, Y±

1 , . . . ,Y±n , Y+

r ∈ (p+)λr , Y−r ∈ p−, such that, for a fixed j

(dαCj )v0((Y

+r )v0,(Y

−l )v0) = δrl (5.2)

Where (Y±r )v0 denotes the image of Y±

r under the projection p±C → p±/g0 = E±v0,C Moreover,

such Y−r are unique, up to a g0C- factor.

We can write the adc1C-invariant splitting

p±C = g0C⊕⊕

λ∈∆±(p±C)

λ

Thus, there is a canonical lift, E±v0,C = p±C/g0C → p±C . Using this lift, we can choose

unique Y−r (Now without the g0C- factor) on the image of this lift.

Now, we take Z ∈ cC. We choose the Cartan subalgebra such that

c⊂ (h⊕I )∩g

, and thus, cC ⊂ (hC⊕IC), that is, Z = Zh+ZI . Where Zh ∈ hC and ZI ⊂ IC. In particular,(Zh)v0 = 0 and ZI commutes with every vector field in gC.


Moreover, as we have seen, g′ is composed by infinitesimal affine transformations (forthe connections ∇ j) and as ∇ j preserves the bundles E±, then so do the vector fields on g′. Thatis if W ∈ g′, then, [W,E±]⊂ E±. Thus, if Z =W + iW ′ ∈ g′C, then

[Z,E±C ] = [W + iW ′,E±+ iE±]⊂ E±+ iE± = E±

C

From this considerations, we have, for Y± complex vector fields tangent to C⊗E±:

dα jC(Y+,adZY−) = Y+(α jC([Z,Y−])− [Z,Y−](αCj (Y

C))−αCj ([Y

+, [Z,Y−]]) (5.3)5.1= −α

Cj ([Y

+, [Z,Y−]]) (5.4)

=−αCj ([Y

−, [Z,Y+]])−αCj ([Z, [Y

+,Y−]]) (5.5)

Now, we write [Y+,Y−] = W++∑l flXl +W−, where W± are complex vector fieldstangent to E±

C . As we have seen, [Zh,E±C ]⊂ E±

C , and thus αCj ([Z,W

±]) = 0. We have

αCj ([Z, [Y

+,Y−]]) = αCj ([Zh+ZI , [Y+,Y−]]) = α

Cj ([Zh, [Y+,Y−]])

= ∑l

αCj ([Zh, flXl]) = ∑

lαCj (Zh( fl)Xl)

= Zh( f j)

That is

dα jC(Y+,adZY−) =−αCj ([Y

−, [Z,Y+]])−αCj ([Z, [Y

+,Y−]]) (5.6)

= dαCj (Y

−, [Z,Y+])−Zh( f j) (5.7)

As we are interested only in the values of dαCj on the point v0, and (Zh)v0 = 0, the last term

vanishes from 5.6 and we obtain:

(dα jC)v0((Y+)v0,(adZY−)v0) =−(dα jC)v0((adZY+)v0 ,(Y

−)v0) (5.8)

Take Y+ = Y+r and Y− = Y−

r choosen as (5.2). Then, as adZY+r = λr(Z)Y+

r we have

(dα jC)v0((Y+r )v0,(adZY−

r )v0) =−(dα jC)v0((adZY+r )v0,(Y

−r )v0)

=−(dα jC)v0((λr(Z)Y+r )v0,(Y

−)v0)

=−λr(Z)((dα jC)v0(Y+r )v0 ,(Y

−)v0) =−λr(Z)

From the uniqueness of Y±r , if we suppose that Z ∈ kerλr, it follows that

adZY−r

−λr(Z)= Y−

r

that is[Z,Y−

r ] =−λr(Z)Y−r


By continuity, when Z ∈ kerλr we have [Z,Y−r ] = 0 =−λr(Z)Y−

r

and therefore −λ ∈ ∆−

Corollary 5.1.10. The subalgebra g0 is the reductive component of the Lie algebras p±

Proof. This is a consequence of the following theorem

Theorem [Bourbaki,(BOURBAKI, 2005), VIII.§3.4 Proposition 13]

Let (g,h) be a splitting Lie algebra and p = h+ gP be a parabolic subalgebra of (g,h).Then, h+gP∩(−P) is the reductive component of p.

In fact, as the Cartan subalgebra cC is contained in g0C = g∆+∩∆−, the above theorem

shows that g0C is the reductive component of p±C and therefore g0 is the reductive component ofp±.

5.2 Technical preparationOn this section we develop a technical result we will use latter on.

Consider a H ′-invariant decomposition E+v0=⊕

i∈J+ E+v0,i and define for every 1 ≤ j ≤ k

E−v0,i, j := Z ∈ E−

v0; dα j(Z,E+

v0,l) = 0 ; ∀l ∈ J+∖i.

We have, for every 1 ≤ j ≤ k,E−

v0=⊕i∈J+

E−v0,i, j

This is a consequence of the following lemma

Lemma 5.2.1. Let E+ and E− be finite dimensional real vector spaces and B : E+×E− → Rbe a non degenerate bilinear map. Assume that E+ admits a decomposition:

E+ =⊕i∈I

E+i

and defineE−

i := Y ∈ E− ; B(Z,Y ) = 0 ∀Z ∈ E+l ; l = i

Then, E− =⊕

i∈I E−i .

Proof. The proof is by induction on the cardinality |I| of I. The case where |I|= 1 is trivial.

Now, suppose that |I|> 1, and consider the induced bilinear map

B : E+/E+i ×E−/E−

i → R

5.2. Technical preparation 139

Let us show that B is non degenerate.

In fact, let Y− ∈ E−/E−i is such that

B(Z+,Y−

) = 0 ∀Z+ ∈ E+/E+i

It is clear that E+/E+i =

⊕l =i E+

l , and thus we identify Z+ with Z+ ∈⊕

l =i E+l . More-

over, let Y− ∈ E− be any representative of Y−. Any other such representative is of the formY−+Y−

i for some Y−i ∈ E−

i . But, by definition, B(Z+,Y−i ) = 0, and therefore,

B(Z+,Y−) = B(Z+,Y−+E−i ) = B(Z+,Y−) = 0 ∀Z ∈

⊕l =i

E+l

and thus Y− ∈ E−i and Y−

= 0.

A similar argument shows that

B(Z+,Y−

) = 0 ∀Y− ∈ E−/E−i ⇒ Z+

= 0

and thus B is non degenerate. By induction, this means

E−/E−i =

⊕l =i

E−l

. where

E−l := Y ∈ E−/E−

i ; B(Z,Y ) = 0 ∀Z ∈ E+j ; j ∈ I∖i, l

Notice that we can identify E−l = E−

l /E−i . But, E−

l ∩E−i = 0 for i = l and thus we

identify E−l = E−

l . That is

E−/E−i =

⊕l =i

E−l

. and

E− = E−i ⊕

⊕l =i

E−l .

The lemma is proved.

We denote by J = +,−×J+×1, . . . ,k, and fix E+v0,i, j = E+

v0,i. We consider, for i ∈ J,the G′-invariant subbundle E i of T M with fiber Ev0,i. We denote E i the corresponding subbundleon T M, and by si the entropy (Definition 4.2.12) of E i with respect to the Rk action3 included inG′. We set σi =

sidim(Ev0,i)

.

Remark 5.2.2. Notice that for every j, we have a smooth splitting T M = T φ ⊕⊕

i∈J+(E+,i, j ⊕

E−,i, j).3 We defined the entropy with respect to a flow, but the definition extends naturally to a definition of

entropy for a subbundle invariant by an Rk- action.


Lemma 5.2.3. There exists a unique smooth connections ∇ j on M that satisfies

∇jdα j = 0 ; ∇

jαi = 0 ∀i ; ∇

j(E i)⊂ E i ; ∇jT φ ⊂ T φ (5.9)

and for, Zi,Zl sections of E i,E l

∇jZiZ

l = pl([Zi,Zl] (5.10)

∇jXi

Zl = [Xi,Zl]+δi jσiZl (5.11)

Where pl denotes the projection T M → E l .

Proof. Analogous to the proof of Lemma. 4.2.1

Lemma 5.2.4. Let Λi denote the determinant bundle of E i, and, for each j = 1, . . . ,k, let ∇ j bethe induced connection on Λi. Then, for each j, the bundle (Λi,∇ j) is flat.

Proof. The proof follows the same path as the analogous proposition for the determinant bundleΛn(E+)*:

∙ Consider Ωi = Ωi j the curvature form of (Λi,∇ j), and define the linear maps B = Bi j by

Ωi j(u,v) = dα j(Bu,v)

.

∙ Consider a primitive β i of Ωi.

∙ Show that∫

M β i j(X j)dM j = 0 (same proof as Lemma 4.2.11).

∙ Show that for every p ∈ 1, . . . ,n−1 we have (Ωi)p∧dαn−pj = 0 (same proof as Lemma

4.2.13).

∙ Conclude that B is nilpotent (same proof as Theorem 4.2.10).

∙ Prove that B is semisimple (same proof as Lemma 4.2.22).

Lemma 5.2.5. For any Y ∈ h and any i ∈ J, we have Tr|Eiv0(Y ) = 0

Proof. In fact, for a fixed j the determinant bundle (ΛE i,∇ j) is flat4, and so is the associatedbundle (ΛE i,∇ j) on the universal cover M. Thus, there exists a parallel section ζi j on M. Thissection is unique up to a constant.4 Remember that we have omitted the index j on the bundle E i. The most complete notation should be

(ΛEσ i, j,∇ j) for σ ∈ +,− and i ∈ J+

5.3. Decomposing g 141

We define the character χi j : G′ → R* defined by

g*ζi j = χi j(g)ζi j ∀g ∈ G′

and its differential dχi j : g′ → R given by

LY ζi j = dχi j(Y )ζi j

Once again (demonstration of Lemma 4.2.29, formula 4.18) we have that (dχi j)|h′ =

Tr|Eiv0

. As g is semisimple and dχi j : g′ → R is a morphism of Lie algebras, dχi j(g) = 0, then,

for Y ∈ h= h′∩g it follows

0 = dχi j(Y ) = Tr|Eiv0(Y )

5.3 Decomposing g

Let us consider n± the unipotent radical of p±, z0 the center of g0. We define

zE0 = Y ∈ z0;adg(Y ) is elliptic

a= Y ∈ z0;adg(Y ) is hyperbolic

We have the equality z0 = zE0 ⊕a. Let m= [g0,g0]⊕zE

0 , then the Langlands decompositionof p+ is

p+ =m⊕a⊕n+

and g0 =m⊕a.

Theorem 5.3.1. Under the above notations, we have h∩a= 0.

Proof. Let Y ∈ h∩a. As Y ∈ h it induces an decomposition

E+v0=⊕µ∈R

E+v0,µ

of E+v0

induced by the adjoint action of Y on p′+/h′ = p+/h= E+v0. Because Y ∈ a ∈ Z(g0), this

decomposition is H ′e-invariant. Lemma 5.2.5 implies that

Tr|Eµ

v0

(Y ) = µ ·dim(Eµv0) = 0

Thus, adg(Y ) has eigenvalues with zero real part. But Y ∈ a and thus adg(Y ) is hyperbolic.Therefore, Y = 0, as we desired.


Remark 5.3.2. It is known [COLOCAR REFERENCIA] that the orbits of an Anosov action arecylinders and the Anosov action acts on this orbit by translation.

As we have already remarked (Corolary 5.1.7), the Lie algebra g0 is the algebra of thegroup that stabilizes the orbit (by the Anosov action) of eH. Moreover, the (right) action of g0

on G/H commutes with the Anosov action on G/H. This means that on the Anosov orbit of eH

in G/H, g0 must also acts as translations, that is, we have a linear map dπact : g0 → Rk.

Lemma 5.3.3. Under the notations above, the map dπact is a morphism of Lie algebras.

Proof. Let O be the orbit through eH, and GO ⊂ G the stabilizer of ≀. We have a group homo-morphism

GO → Di f f (O)

But as we have seen, GO must acts though translations on O , that is, the image of GO onDi f f (O) is isomorphic to Rk ⊂ and the induced map

πact : GO → Rk

is a Lie group homomorphism. Its differential is precisely the map dπact and it is therefore a Liealgebra homomorphism.

Corollary 5.3.4. The restriction dπact |a : a→ Rk is injective.

Proof. In fact, suppose that we have a vector field Y ∈ g0 such that dπact(Y ) = 0, this meansthat Y acts trivialy on the orbit, and in particular, it fixes the point eH, that is, Y ∈ h. Thus,ker(dπact) = h. From Theorem 5.3.1, a∩ker(dπact) = 0, and thus dπact |a is injective.

Lemma 5.3.5. We have [g0,g0]⊂ h⊂m

Proof. As dπact is a morphism of Lie algebras, it follows that dπact([g0,g0]) = 0 and thus,[g0,g0]⊂ h. Now, let x ∈ h. We consider the decomposition

x = x0 + xc

where x0 ∈ [g0,g0] and xc ∈ zE0 ⊕a= Z(g0).

Consider the adjoint action of xc on g. As xc ∈ h∩Z(g0) it follows that the decompositionof g in eigenspaces is H ′

e invariant. On the other hand, xc ∈ h and thus, from Lemma 5.2.5 itfollows that ad(xc) has no eigenvalue with non zero real part, that is xc is elliptic, that ish⊂ [g0,g0]⊕ zE

0 =m

Remark 5.3.6. As the map dπact : g0 →Rk is clearly surjective, it follows that there exists somesubspace L′ ⊂ zE

0 such that the abelian subalgebra L = L′⊕a is k dimensional and such that theanosov action is given by right multiplication of the elements of L.


Proposition 5.3.7. Consider the projection π : G×Rk → G and the holonomy representationρ : π1(M)→ G′ = G×Rk. Suppose that the Anosov action is faithfull. Then, Γ = π ∘ρ(π1(M))

is discrete.

Proof. Let us denote Γ =(π ∘ρ(π1(M))

)e, where ( · )e denotes the identity component of the

closure on G.

Suppose that Γ is non trivial.

It is known (BARBOT; MAQUERA, 2011) that an Anosov action admits only a finitenumber of compact orbits of fixed volume. As G acts by isometries, for a fixed compact orbit O ,sufficiently small elements of G (that is, elements on a neighborhhod UO ⊂ G of the identity)must send this orbit to itself. As Γ is connected, we have that UO ∩ Γ generates Γ and thus Γ

must stabilizes O .

It is known that if G is a real semisimple Lie group and P is a parabolic subgroup then P

is closed in G, in particular, if P± are oposed parabolic subgroups, then P+∩P− is closed in G. Ifwe take the parabolic subgroups P± associated with the Lie subalgebras p±, then GO = P+∩P−

has Lie algebra g0 and thus, is the stabilizer of an orbit. Therefore, G/GO is the orbit space ofthe Anosov action (lifted to its universal cover), and it has a manifold structure.

As Γ stabilizes the compact orbits, which are dense in G/GO , it follows that Γ stabilizesevery orbit.

Then

Γ ⊂⋂

g∈G

gGOg−1

Let us denote N =⋂

g∈G gGOg−1. Notice that N ⊂ GO and, moreover, N is a normalsubgroup of G. As G is semisimple, then N is also semisimple, and thus, N is on the kernel ofthe projection map

πact : GO → Rk

This implies that N ⊂ H. The same arguments shows that N belongs to the kernel of theprojections maps

πact : gGOg−1 → Rk

for every g, that is N ⊂ gHg−1 for every g. Therefore the elements of Γ acts trivially on G/H.


Now, we observe that the projection of ρ(Γ) on the second coordinate

Γρ→ G×Rk → Rk

acts on G/H as an element of the Anosov action, and thus, let γ ∈ Γ such that ρ(γ) = (γ1,γ2),γ1 ∈ Γ ⊂ G, then γ1 acts trivially on G/H, and thus, γ acts on G/H precisely as an element ofthe anosov action, that is, if we consider φ the lift of the anosov action to the universal coverM = G/H, then, there exists t(γ) ∈ Rk such that for every m ∈ M

γ · m = φt(γ)(m)

This means that on the manifold M = Γ∖M, the element γ(t) ∈ Rk acts trivialy, whichcontradicts the faithfulness of the action.

Theorem 5.3.8. Under the hypothesis of Theorem 1.0.1, let (G,K,Γ,a) be our quasi-algebraicmodel. Suppose that G is an algebraic group. Then a contained in the Cartan subspace of G

Lemma 5.3.9. If G is algebraic we have m⊂ h

Proof. As dπact is a morphism of Lie algebras, dπact([g0,g0]) = 0, and thus, [g0,g0] ⊂ h =

ker(dπact). It remains to show that zE0 ⊂ h.

Take ξ0 ∈ zE0 . As ad(ξ0) is elliptic, its action on gC = g⊕ ig has pure imaginary eigen-

values. If we denote GC the complexification of the group G, we have

exp(t ad(x0)) ⊂ ad(GC) is compact

As ξ0 ∈ g, then

exp(t ad(x0)) ⊂ ad(G)⊂ ad(GC)

and thus,

exp(adzE0 ) ⊂ ad(G) is compact

.

Let us denote L = exp(adzE0 ). As L is compact on ad(G) it has only compact orbits,

however, zE0 ⊂ g0 and thus, L stabilizes the orbits of the Anosov action.

The following lemma finishes the proof.

Lemma 5.3.10. Suppose that the Anosov action is faithful. Let O ⊂ G/H be an orbit of theAnosov action and GO ⊂ G be the stabilizer of O . Suppose that K ⊂ GO is compact. Then,K ⊂ H.


Proof. In fact, as GO is the stabilizer of an orbit, and G acts transitively, then the orbits of theright action

G/H y GO

coincide with the orbits of the Rk-action.

As the action is faithful we have the identification

GO/H = Rk

As K is compact, the projection

K → GO/H

is trivial, and thus K ⊂ H

Theorem 5.3.11. Under the hypothesis of Theorem 1.0.1, let (G,H,Γ,a) be our quasi-algebraicmodel. Suppose that H is a compact subgroup, then the action is the Weyl chamber action.

Proof. First, let us show that m= h.

Let us show that m⊂ h (we have already proved that h⊂m). Let ze = h∩m, it preservesa Riemannian metric on the Lie algebra g, in fact the metrics preserved by it forms an convexcone on the space of metrics on g. The group H acts on this convex cone, and as H is compact,it fixes a point((HORNE, 1978), Theorem 5.1). This metric is therefore invariant by h. It cantherefore be extended by left translation to G/H. This metric is also preserved by π1(M) forH1(π1(M),R) = 0.

Now, suppose that there exists some element L ∈m∩hc. Then L must acts via isometries(for they commute with ze), which is not possible due to the Anosovness property.

The compacity of H implies that h can not contain any other hyperbolic element and,therefore, a is a maximal abelian subalgebra of hyperbolic elements of g, that is, it is the Cartansubspace of g.

Finally, π1(M) is a lattice, for M = Γ∖G/H is compact, and thus, Γ∖G is also compact,for it is a fibration over M with compact fiber H.

147

CHAPTER

6CONCLUSION

We have obtained in this work a result similar to the one given by Benoist, Foulon andLabourie (BFL - (BENOIST; FOULON; LABOURIE, 1992)). It is not, however, in the strictestsense, a generalization of the result given by BFL, for the simple reason that in the case k = 1 ourresult is actually weaker for we conclude only the quasi algebricity of the action, whereas BFLconcludes it’s algebricity. Moreover, our result requires some additional hypothesis. Nonetheless,we feel that our result may represents a important step in the direction of proving the generalconjecture that Anosov actions are in fact algebraic.

This feeling is powered by Klein’s views on geometry, that is, that geometry is the studyof things which are invariant by the action of a certain group. According to this view, algebraicAnosov actions should be associated with some kind of geometric structure.

On this work, we have shown that (under some mild hypothesis) for a specific geometricstructure, that of generalized k contact, the associated Anosov action is quasi-algebraic. It isour hope that a similar strategy could be used to prove the same kind of result for more generalgeometric structures.

We devote the following sections to a brief study of the hypothesis on our theorem andsome possible avenues for future research.

6.1 Theorem 4.0.1 hypothesis

On this section we shall make some brief comments on the importance and reasonabilityof the hypothesis used on Theorem 4.0.1, explicitly, the smoothness of the invariant bundles,the faithfulness of the Anosov action and the nullity of the first cohomology of the fundamentalgroup of the base manifold.

∙ Smoothness

148 Chapter 6. Conclusion

The smoothness of the invariant bundles was used to prove that the adapted connection (Lemma4.2.1) was smooth (Remark 4.2.2). This smoothness was of crucial importance on some of ourcalculations, for example, the results about killing vector fields and compatible connections onsubsection 2.3.3 require the smoothness of the connection.

It is known that algebraic Anosov actions have smooth invariant bundles, and therefore,the smoothness hypothesis is in accordance with the standing conjecture that every Anosovaction of higher rank is algebraic. This means that should the conjecture be proven true, thishypothesis is not actually necessary. However, we must proceed with caution, it has been provedby Foulon, P. and Hasselblatt, B. ((FOULON; HASSELBLATT, 2013)) that the smoothness isa necessary condition for the case of flows, that is, they had shown that there exists a contactAnosov flow (on dimension 3) without smooth invariant bundles, which is not algebraic.

∙ Faithfulness and H1(π1(M),R) = 0

Faithfulness is an irreducibility hypothesis, if the action is not faithful, than, the basemanifold M is a fiber bundle M → B whose standard fiber is a torus Tl and there is a Rk−l actiontransverse to the fibres, which induces a faithful action on B. This induced action is Anosov if,and only if the original action is Anosov, moreover, it is easy to see that, a generalized k-contactstructure compatible with the Anosov action on M induces a generalized (k− l)-contact structureon B which is compatible with the induced action. This means that there is a canonical way tobuild a faithful action from a non faithful one.

The cohomology condition is more subtle, it is a technical condition, and it seems thatthere is no reason to suppose that a manifold supporting an Anosov action should satisfy thiscondition. In fact, this condition was not used to prove the quasi-algebricity, but to prove thatthe fundamental group π1(M), represented as a discrete subgroup of G′ = G×Rk, can actuallybe represented as a discrete subgroup of G, this allowed us to reduce our (G′,X)-structure to a(G,X)-structure, where G is a semisimple Lie group.

On BFL’s work, they did not use this hypothesis, but the flexibility of dimension oneallowed them to re-parametrize the flow in such a way that π1(M) acts trivially on the abelianfactor of G′ = G×R, but the rigidity of higher ranked actions means that we could not re-parametrize in our case.

6.2 Further work

Under very mild conditions, we have obtained that quasi-algebricity of the action, it istherefore, a rather natural to question

Question 1: Under which conditions a quasi-algebraic action is an algebraic action?

6.2. Further work 149

This is an open question, a purely algebraic one, and a rather difficult problem. It is related toanother active area of research that is the question of when does an homogeneous space G/H

admits a compact quotient Γ∖G/H. This is a well understood problem when H is compact, butfor general closed H this is an open problem. A small survey on this area was given by F. Kasselon (KASSEL, 2009). Given the difficulty of this question, let us pose some similar questionassuming stronger hypothesis.

Under a few extra hypothesis (faithfulness of the action and nullity of the first cohomologygroup of π1(M)), we obtained a reduction of the quasi-algebraic action to a semisimple Liegroup. Moreover, at least part of the action is given by right translation of an abelian subalgebraof hyperbolic elements 5.3.6. This raises the following question.

Question 2: Which additional dynamical or topological hypothesis on M can be used to removethe algebricity hypothesis from Theorem 5.3.8? And to remove the compact hypothesisfrom Theorem 5.3.11

Another possibility for further research is to try to generalize the proof for nilpotentgroup actions. More precisely

Question 3: Consider an Anosov action of a nilpotent Lie group associated with a transversepluricontact structure (Definition 3.2.50). is this action quasi algebraic?

On another direction, we can pose questions when we have prior knowledge about theambient manifold

Question 4: Which algebraic Anosov action admits a compatible generalized k-structure, ormore generally, a transverse pluricontact structure?

There are also open questions about the geometric structure we have defined

Question 5: Does the generalized k-contact structure admits some kind of Darboux theorem?

This question is motivated by the fact that there is a Darboux theorem for both contact pairs andfor k-contact structures. Moreover, consider R2n+k with coordinates (z1, . . . ,zk,x1,y1, . . . ,xn,yn),then it is easy to see that for any symplectic matrix S1, . . . ,Sk, Sl = (sl

i j)i j, the 1-forms

dzl + ∑1≤i< j≤n

sli jxidy j l = 1, . . . ,k

define a generalized k-contact structure on R2n+k with splitting Rk ×R2n. It is natural to questionif every generalized k-contact structure is locally of this form.

Finally, another possibility for new investigations is inspired by the following two resultsby Y. Fang:

150 Chapter 6. Conclusion

Theorem 6.2.1 (Theorem 2: (FANG, 2003)). Anosov flows on dimension 5 with smooth invariantbundles which preserves a non degenerated pseudo-metric are (smoothly conjugated to) algebraic.

Theorem 6.2.2 (Theorem 1: (FANG, 2007)). Uniformly quasi-conformal Anosov flows ofco-dimension at least 2 which preserves a volume form and such that E+⊕E− is smooth are(smoothly conjugated to) algebraic.

Another such result also proved by Y. Fang shows that the uniformly quasi-conformalAnosov flows of co-dimension at least 2 such that E+⊕E− is smooth than, is is algebraic.

On the heart of the proof of both of this theorems lies BFL’s result. That is, Y. Fang usesthe imposed dynamical conditions to, at least on some cases, recover the hypothesis of BFL’stheorem. This motivate us to believe that it is possible to give generalizations for both of thistheorems.

151

APPENDIX

A(G,X)-STRUCTURES

A.0.1 Geometric structures

Definition A.0.1. Consider a category C such that every morphism is an isomorphisms. Suchcategory is called a groupoid. In set theoretical terms, a groupoid is a set G with a "sort of"binary operation that is not actually defined for every pair g,h ∈ G .

To translate from the categorical view to the set theoretical view, we take the set G asthe set of all morphisms of the category C . If C has only one object, the groupoid we obtainis actually the group of every morphism of this object. If C has more then one object, then notevery morphism can be composed, but when the composition is defined, it is associative andalways have a unique unit.

The most common example of groupoid (and the one we are interested in) arise fromlocal isomorphisms. Consider a category E and an object X of E . Consider a subcategory C ofsub-objects of X (that is monomorphisms 1 Y → X) and all isomorphisms between them. C is bydefinition a groupoid. In our case, we want E the category of topological spaces, X a topologicalmanifold, and GX the category whose objects are open sets of X and whose morphisms arehomeomorphisms between then. We call GX the groupoid of local homeomorphims of X . Thisgrupoid has an additional structure which makes it into a Pseudo-group, that is, it is endowedwith a restriction map.

Definition A.0.2. Let X be a topological space. Then, a pseudo-group G in X will mean subcat-egory of the groupoid GX such that

1 a morphisms f is a monomorphisms if for every pair of morphisms g1,g2 we have

f ∘g1 = f ∘g2 ⇒ g1 = g2

152 APPENDIX A. (G,X)-structures

∙ The objects of G cover X

∙ For every local homeomorphism g in G , its restriction to any open set of X is also anelement of G

∙ If g is an element of G , then so is its inverse.

∙ Consider a family Ui of open sets and a homeomorphism g : U = ∪Ui →V ⊂ X suchthat every g|Ui is an element of G , than so is g.

The pseudo-groups GX can be used to define manifolds, and more generally (we shallbellow), geometric structures on the manifolds.

Definition A.0.3 (G -Manifold). Consider G a pseudo-group in Rn, a G -manifold is a topological,Hausdorff space M, and an atlas Ui,ϕi : Ui →Vi ⊂ Rn, such that the transition maps ϕi ∘ϕ

−1j

are in G .

If we take G the pseudo-group of local diffeomorphisms of class Ck, k = 1, . . . ,∞,ω , weobtain a smooth manifold in the traditional sense.

Definition A.0.4 ((G ,X)-Manifold). We take X to be an arbitrary manifold and G a pseudo-group on X . A G ,X)-manifold M is a topological Hausdorff space M with an atlas with valueson X such that the transition maps are in G . A (G ,X) structure on M is a maximal G -atlas.

Thurston proved that if G acts transitively2, then this definition does not give rises tonew types of manifolds, however, such structures can reveal additional geometric properties ofthe manifold. For example

Example A.0.5. Consider Rn = Rk ×Rn−k, and consider local diffeomorphisms ϕ of Rn of theform

ϕ = (ϕ1(x,y),ϕ2(x))

This is a pseudo-group on Rn and a manifold with this kind of structure, is actually a foliation ofco-dimension k

Definition A.0.6. A map ξ : M → N between (G ,X) manifolds is called a (G ,X) map if, ξ iscontinuous and for each charts ϕ : U → X and ψ : V → X of M and N respectively, we have

ψξ ϕ−1 : ϕ(U ∩ξ

−1(V ))→ ψ(ξ (U)∩V )

is an element of G .

Obs.: A.0.7. A function ξ : M → N between (G ,X) manifolds is a (G ,X)-map if, and only if,for each point u ∈ M there exists a chart ϕ : U → X , U ∋ x, such that ξ |U is an homeomorphismonto it’s image, and ϕξ−1 : ξ (U)→ X is a chart of N.2 For every pair of points in M, there exists a local homeomorphism in g that takes one point to another

153

Definition A.0.8. A pseudo-group G on X is said to acts analytically on X if every local homeo-morphisms defined on a connected open set, can be extended to the whole connected componentof it’s domain, and moreover, this extension is unique, that is, if two local homeomorphismthat coincide on a open set, then, they coincides on the connected component of this set. Imparticular, if X is connected, then the pseudo-group G is actually a group G acting on X , and thepseudo-group can be recovered from he restrictions of G to open subsets of X .

A.0.2 Developing map

The goal of this section is to prove the following proposition

Proposition A.0.9. Consider X a connected topological space and G a group acting analyticallyon X . Consider M a (G,X) manifold. Then, for every simple connected open subset U of X ,there exists a (G,X) map from θ : U → X , moreover, any other such map is of the form g∘θ

where g denotes an element of G.

Lemma A.0.10. Consider N a connected (G,X) manifold, and (G,X) maps f , f ′ : N → X . If f

coincides with f ′ on a open, non empty, set, then they are are equal.

Proof. Consider

W = p ∈ N ; ∃Up ⊂ N f (q) = f ′(q) ∀q ∈Up

where Up denotes some open neighbourhood of p. By hypothesis, this set is non empty, and bydefinition, it is open. Let us show that it is closed.

Let w be an accumulation point in the closure W . And let ϕ : Uw → X be a (G,X)-chartaround w. By definition of (G,X)-map, the compositions f ∘ϕ−1 and f ′∘ϕ are given by elementsg and g′ of G. Moreover, as w is an accumulation point, there exists w′ ∈Uw ∩W , and thus, thereexists an open set w′ ∈V ⊂Uw such that f and f ′ coincides on V . Therefore, g and g′ coincideson ϕ(V ).

As G acts analytically on X , this means that g = g′ on ϕ(Uw) and thus f = f ′ on Uw.That is w ∈W .

Proof of Proposition A.0.9. Le us fix a point v0 ∈ U , and ϕ0 : V0 → X a chart around v0. Weshall define the developing map θ : U → X in such a way that θ|V0

= ϕ0.

We consider local charts ϕl : Ul → XL∈Z+ such that U = ∪lUl and Ul ∩Ul+1 = forevery l.

The definition of (G,X)-structure implies that for every l = 0 there exists gl ∈ G suchthat ϕl−1 = gl ∘ϕl .

154 APPENDIX A. (G,X)-structures

We define, for v ∈Vl

θ(v) = g1 ∘ · · · ∘gl ∘ϕl(v)

Let’s see that this application is well defined. We have gl+1 ∘ · · · ∘gl+k ∘ϕl+k = ϕl andthus, if v ∈Vl ∩Vl+k, it follows that

g1 ∘ · · · ∘gl ∘ϕl(v) = g1 ∘ · · · ∘gl+k ∘ϕl+k(v)

as we wanted.

The map θ we defined is of course a (G,X) map, and the previous lemma (LemmaA.0.10), shows that the prolongation θ is unique.

Moreover, suppose that θ and θ ′ are two possible developing maps, then, for a fixedv0 ∈U , there exists g such that θ = g∘θ ′ on a neighbourhood of v0. The Lemma A.0.10 impliesthat θ = g∘θ ′ on U .

155

APPENDIX

BCONNECTIONS ON VECTOR BUNDLES

Definition B.0.1. Let E → M be a vector bundle with rank r. We shall denote by A p(E) thespace of p-forms on M with values on E, that is, a section of the tensor product (of vector bundleswith the same base space) E ⊗ΛpT *M. for convenience, we set A 0(E) the space of sections ofE. A connection on E is a homomorphism

∇ : A 0(E)−→ A 1(E)

such that

∇( f ζ ) = ζ d f + f ∇ζ

Let s = (s1, . . . ,sr) be a local frame field of E over an open subset U ⊂ M, that is,

1. si ∈ A 0(E|U ), for i = 1, . . . ,r

2. (s1(x), . . . ,sr(x)) is an ordered basis of Ex, for each x ∈U

Thus, given a connection ∇ on E, we can write

∇si = ∑j

s jωj

i ; ωj

i ∈ A 1|U

The matrix of 1-forms ω = (ωj

i )i j is called the connection form of ∇ with respect to theframe s. Making use of the matrix notation we write:

∇s = s ·w

156 APPENDIX B. Connections on vector bundles

Therefore, if we set ζ = ∑ζ isi, ζ i ∈ A 0U

, an arbitrary section of E over U , then

∇ζ = ∑i

∇ζisi = ∑

i

(sidζ

i +ζi∇si)

(B.1)

= ∑i

(sidζ

i +ζi∑

js jω

ji)

(B.2)

= ∑j

s j(dζ

j +∑i

ζiω

ji)= dζ +ωζ (B.3)

We call ∇ζ the covariant derivative of ζ . We can evaluate this at a tangent vector X of M

and we obtain:∇X ζ := (∇ζ )(X)

If we take another local frame s = (s1, . . . , sr) of U , they are related by

s = s ·a

where a : U → GL(n,R). If ω is the connection form with respect to s, we have:

sω = ∇s = ∇(s ·a) = ∇(s)a+ s ·da

= (sω)a+ sda = s(ωa+da) = s(a−1ωa+a−1da)

There exists a unique way to extend this connection, to a linear map

∇ : A p(E)→ A p+1(E) ; p ≥ 0

that satisfies the Leibniz condition

∇(σ ·ϕ) = (∇σ)∧ϕ +σ ·dϕ ; σ ∈ A 0(E) ϕ ∈ A p

The above formula is precisely the way to define this extension. Now, using this extensionwe define

R = ∇∘∇A 0(E)→ A 2(E)

R is called the curvature of the bundle E. Note that

∇2( f ζ ) = ∇(ζ d f + f ∇ζ ) = (∇ζ )∧d f +ζ d2 f +d f ∧∇ζ + f ∇

2ζ = f ∇

2ζ

thus ∇2 is actually A 0-linear, thus, R is actually a 2−form with values on End(E), if we onceagain, use the matrix notation, the curvature form Ω with respect to a local frame s, is defined by

sΩ = ∇2s

Thus

sΩ = ∇(∇s) = ∇(sω) = (∇s)∧ω + sdω = sω ∧ω + sdω

157

that isΩ = dω +ω ∧ω

Chosing another local frame s, the corresponding curvature form Ω is related to Ω by

Ω = a−1Ωa

In fact:

sΩ = ∇2s = ∇

2(sa) = ∇((∇s) ·a+ s ·da

)= ∇

2s ·a− (∇s)∧da+(∇s)∧da+ s ·d2a = sΩa = sa−1Ωa

B.0.1 Connections on Associated Vector Bundles

Let E be a vector bundle over M, let E* denote the dual vector bundle, bilinear mapinduced by the duality paring

⟨ · , · ⟩ : E*x ×Ex → R

induces a dual pairing⟨ · , · ⟩ : A 0(E*)×A 0(E)→ A 0

We can use this induced pairing to define a connection on E*, also called ∇, by thefollowing formula

d⟨ξ ,η⟩= ⟨∇ξ ,η⟩+ ⟨ξ ,∇η⟩ ; ξ ∈ A 0(E) η ∈ A 0(E*)

Let’s see how this relates to the connection form and the curvature. If s = (s1, . . . ,sr) is alocal frame of E, let t = (t1, . . . , tr) a local frame of E* that is dual to s, that is

⟨t i,s j⟩= δi j or ⟨t,s⟩= Ir

where s is a row vector and t is a column vector. If ω = (ω ij)i j is the matrix of the connection

form, then:

0 = dδi j = d⟨t i,s j⟩= ⟨∇t i,s j⟩+ ⟨t i,∇s j⟩= ⟨∇t i,s j⟩+ ⟨t i,∑k

skωkj ⟩= ⟨∇t i,s j⟩+ω

ij

that is,∇t i =−∑

jω

ijt

j or ∇t =−ωt

Thus, if η = ∑ηit i is an arbitrary section of E*, we have (following the same calculations of(B.1)):

∇η = dη −ηω

Analogous calculations also show that, if Ω is the curvature form of E with respect to a localframe s, then, with respect to the dual frame t, we have

∇2t =−Ωt


To summarize, if s is a local frame on E, and t is the dual frame on E*, if ω ij is the

connection form of E with respect to s and Ω is the curvature form, we have:

∇s = s ·ω ; ∇2s = sΩ

∇t =−ω · t ; ∇2t =−Ωt

Now we consider two vector bundles E,F over the same base space M, with connections∇E and ∇F , then we can define natural connections ∇E ⊕∇F on the Whitney sum E ⊕F and∇E⊗F on the tensor product E ⊗F . The former is trivial and the latter is given by

∇E⊗F = ∇

E ⊗ IF + IE ⊗∇F

It is easy to see that the associated curvatures are

RE⊕F = RE ⊕RF

RE⊗F = RE ⊗ IF + IE ⊗RF

If we chose local frames s = (s1, . . . ,sr) and t = (t1, . . . , tl) of E and F , and denote byωE ,ωF ,ΩE ,ΩF the connection and curvature forms of E and F with respect to this local frames,we have

ωE⊕F =

(ωE 00 ωF

)and ΩE⊕F =

(ΩE 00 ΩF

)also

ωE⊗F = ωE ⊗ IF + IE ⊗ωF and ΩE⊗F = ΩE ⊗ IF + IE ⊗ΩF

It is clear that this construction generalizes to finite tensor products and finite sums (or afinite combination of both)1. For example, lets consider E ⊗F ⊗W , with local frames s, t and u.

∇E⊗F⊗W = ∇

E ⊗ IF⊗W + IE ⊗∇F⊗W

= ∇E ⊗ IF ⊗ IW + IE ⊗∇

F ⊗ IW + IE ⊗ IF ⊗∇W

An interesting case is E⊗p, on which we have the connection

∇E⊗p

=p

∑j=1

I⊗ j−1E ⊗∇

E ⊗ I⊗(p− j)E

1 It is of special interest to consider the case End(E) = E ⊗E*. Consider a local frame field s on E andit’s dual t on E*. Let ξ be End(E) valued p-form, that is

ξ = ∑ξijsi ⊗ t j ; ξ

ij ∈ A p

159

Let’s remember that the exterior power Λk(E) is obtained by taking the quotient of E⊗k

by the two sided ideal I given by

I = x1 ⊗·· ·⊗ xk ∈ E⊗k ; xi = x j for some i = j

With those tools at hand, we can calculate the connection of the exterior power ΛkE. If we chosea local frame s

∇ΛkE(si1 ∧·· ·∧ sik) = ∑

j(−1) j+1si1 ∧·· ·∧∇

Esi j ∧·· ·∧ sik

Now, if, we suppose that k is maximal, that is k = rank(E) = r, then, given a local frames, the only element on ΛkE modulo A 0 is s1 ∧·· ·∧ sr. Also we have

s1 ∧·· ·∧∇Es j ∧·· ·∧ sr = s1 ∧·· ·∧ (∑

iω

ijsi)∧·· ·∧ sr

= (−1) j−1ω

jj s1 ∧·· ·∧ sr

thus

∇ΛrE = ∑

jω

jj s1 ∧·· ·∧ sr

that is, the connection form on ΛrE is the trace of the original connection form on E

ωΛrE = Tr(ωE)

Then

∇ξ = ∑i j

∇(ξ

ijsi ⊗ t j)= ∑

i j

(((∇si)⊗ t j + si ⊗ (∇t j))∧ξ

ij +dξ

ij(si ⊗ t j)

)= ∑

i j

((−1)p

ξij ∧ ((∇si)⊗ t j + si ⊗ (∇t j))+dξ

ij(si ⊗ t j)

)= ∑

i j

(dξ

ij(si ⊗ t j)+(−1)p

ξij ∧ ((∇si)⊗ t j + si ⊗ (∇t j))

)= ∑

i j

(dξ

ij(si ⊗ t j)+(−1)p

ξij ∧ ((∑

kskω

ki )⊗ t j + si ⊗ (−∑

lω

jl t l))

)= ∑

i j

(dξ

ij +(−1)p

∑k

(ξ

kj ∧ω

ik −ξ

ik ∧ω

kj))

si ⊗ t j

The curvature R of ∇ on E is in fact a End(E) valued 2-form and we write,

R = ∑Ωijsi ⊗ t j


On the other hand, as Ω = dω +ω ∧ω we have:

ΩΛrE = dω

ΛrE +ωΛrE ∧ω

ΛrE

= d(Tr(ωE +Tr(ωE)∧Tr(ωE)

= Tr(dωE)+Tr(ωE ∧ω

E) = Tr(ΩE)

Now, a local frame s of ΛrE is just a volume form, that is, we can choose s = (s1, . . . ,sr)

is a local frame of E such that s = f ·det(s) = f · s1 ∧·· ·∧ sr, thus

sΩΛrE = Tr(sΩ

E)

In particular, if E is a integrable sub-bundle of T M, we have

ΩΛrE*

= Tr(ΩE*) = Tr(ΩT *M

|E ) (B.4)

161

BIBLIOGRAPHY

ANOSOV., D. V. Geodesic flows on closed riemannian manifolds of negative curvature. TrudyMat. Inst. Steklov., v. 90, p. 3 – 210, 1967. Citation on page 17.

APOSTOLOV, V.; CALDERBANK, D. M. J.; GAUDUCHON, P.; LEGENDRE, E. Toric contactgeometry in arbitrary codimension. pre-print, p. 1–22, 08 2017. Citations on pages 41, 54, 56,and 57.

ARBIETO, A.; MORALES, C. Dynamics of Partial Actions. Rio de Janeiro, RJ: IMPA, 2009.Citation on page 40.

BANDE, G.; HADJAR, A. Contact pairs. Tohoku Math. J. (2), v. 57, p. 247 – 260, 2005.Citations on pages 41 and 49.

BARBOT, T.; MAQUERA, C. Transitivity of codimension-one anosov actions of rk on closedmanifolds. Ergodic Theory and Dynamical Systems, v. 31, p. 1–22, 2011. Citations on pages18 and 143.

. Algebraic anosov actions of nilpotent lie groups. Topology and its Applications, v. 160,n. 1, p. 199 – 219, 2013. Citations on pages 40 and 60.

BARTHELMé, T.; BONATTI, C.; GOGOLEV, A.; R., H. F. Anomalous anosov flows revisited.pre-print, p. 1–26. Available: <https://arxiv.org/pdf/1712.07755.pdf>. Citations on pages 11,13, and 17.

BENOIST, Y. Orbites des structures rigides (d’aprs m. gromov), integrable systems and foliations.Progress in Math, v. 145, p. 1–17, 1997. Citation on page 22.

BENOIST, Y.; FOULON, P.; LABOURIE, F. Flots d’anosov a distributions stable et instabledifferentiables. Journal of the American Mathematical Society, v. 5, n. 1, 1992. Citations onpages 11, 13, 18, 19, 90, and 147.

BOLLE, P. A contact condition for coisotropic manifolds of a symplectic manifold. ComptesRendus de l’Académie des Sciences. Série I, p. 211–230, 01 1996. Citations on pages 41, 49,50, and 55.

BOURBAKI, N. Elements of Mathematics: Lie Groups and Lie Algebras, Chapters 7-9.[S.l.]: Springer, 2005. Citations on pages 113 and 138.

CANTRIJN, F.; IBROT, A.; LéON, M. de. On the geometry of multisymplectic manifolds. J.Austral. Math. Soc. (Series A), v. 66, p. 303–330, 1999. Citation on page 51.

CHOI, W.; PONGE, R. Privileged coordinates and nilpotent approximation of carnot manifolds,i. general results. pre-print, p. 1–40, 09 2017. Available: <arXiv:1709.09045>. Citation onpage 54.

DöRNER, M.; GEIGES, H.; ZEHMISCH, K. Finsler geodesics, periodic reeb orbits, and openbooks. European Journal of Mathematics, v. 3, 11 2016. Citation on page 67.

https://arxiv.org/pdf/1712.07755.pdf

arXiv:1709.09045

162 Bibliography

ERP, E. V. Contact structures of arbitrary codimension and idempotents in the heisenberg algebra.arXiv:1001.5426, p. 1–15, 01 2010. Citations on pages 41, 54, and 55.

FANG, Y. Geometric anosov flows of dimension 5. Comptes Rendus Mathematique, v. 336,n. 5, p. 419 – 422, 2003. Citation on page 150.

. On the rigidity of quasiconformal anosov flows. Ergodic Theory and Dynamical Sys-tems, v. 27, n. 6, p. 1773–1802, 2007. Citation on page 150.

FERES, R. Rigid geometric structures and actions of semisimple lie groups. Rigidité, groupefondamental et dynamique, Soc. Math. France, Paris, v. 13, p. 121–167, 2002. Citation onpage 22.

FOULON, P.; HASSELBLATT, B. Contact anosov flows on hyperbolic 3–manifolds. Geom.Topol., v. 17, n. 2, p. 1225–1252, 2013. Citations on pages 18 and 148.

FOULON, P.; LABOURIE, F. Flots d’anosov a distributions de lyapounov differentiables. C. R.Acad. Sci. Paris, v. 309, p. 255–260, 1989. Citation on page 111.

FRANKS, J. Anosov diffeomorphisms. Global Analysis (Proc. Symp. Pure Math.), Amer.Math. Soc., p. 61 – 93, 1970. Citation on page 17.

GHYS., E. Codimension one anosov flows and suspensions. Lecture Notes in Math., n. 1331,p. 59 – 72, 1988. Citation on page 18.

GROMOV, M. Rigid transformation groups. Géometrie différentielle, Ed. Travaux encours,Paris, v. 33, p. 65–139, 1988. Citation on page 29.

HANDEL, M.; THURSTON, W. P. Anosov flows on new three manifolds. Inventiones mathe-maticae, v. 59, n. 2, p. 95–103, Jun 1980. Citations on pages 11, 13, and 17.

HELGASON, S. Differential Geometry, Lie Groups, and Symmetric Spaces. [S.l.]: ElsevierScience, 1979. (Pure and Applied Mathematics). Citations on pages 81 and 109.

HIRSCH, M. W.; PUGH, C. C.; SHUB, M. Invariant manifolds. Bull. Amer. Math. Soc., v. 76,n. 5, p. 1015–1019, 09 1970. Citation on page 40.

HOF, H. C. I. An anosov action on the bundle of weyl chambers. Ergodic Theory and Dynam-ical Systems, v. 5, p. 587–593, 1985. Citation on page 80.

HORNE, J. On the automorphism group of a cone. Linear Algebra and its Applications, v. 21,n. 2, p. 111 – 121, 1978. Citation on page 145.

KAMMEYER, H. An explicit rational structure for real semisimple lie algebras. Journal of LieTheory, v. 24, n. 2, p. 307–319, 2014. Citations on pages 80, 81, and 83.

KASSEL, F. Quotients compacts d’espaces homogènes réels ou p-adiques. Phd Thesis (PhDThesis) — Département de Mathématiques d’Orsay - Université Paris-Sud 11, 2009. Citationon page 149.

KATOK, A.; HASSELBLATT, B. Introduction to the Modern Theory of Dynamical Systems.[S.l.]: Cambridge University Press, 1997. (Encyclopedia of Mathematics and its Applications).Citation on page 67.

Bibliography 163

KOBAYASHI, S.; NOMIZU, K. Foundations of differential geometry vol 1. [S.l.]: Wiley,1963. Citations on pages 35 and 43.

MONTANO, B. C. Integral submanifolds of r-contact manifolds. Demonstratio Mathematica,v. 41, n. 1, p. 189–202, 2017. Citations on pages 41 and 51.

MOSTOW, G. D. The extensibility of local lie groups of transformations and groups on surfaces.The Annals of Mathematics, v. 52, p. 606–636, 1950. Citation on page 129.

NESTEROV, A. I. Principal g-bundles. Nonassociative algebras and its applications: thefourth international conference, Marcel Dekker Inc, p. 247–257, 2000. Citation on page 43.

NEWHOUSE, S. On codimension one anosov diffeomorphisms. American Journal of Mathe-matics, v. 92, p. 761–770, 1970. Citation on page 17.

PATRãO, M.; SANTOS, L.; SECO, L. A note on the jordan decomposition. Proyecciones, v. 30,08 2008. Citation on page 109.

PLANTE, J. F. Anosov flows, transversely affine foliations, and a conjecture of verjovsky.Journal of the London Mathematical Society, s2-23, n. 2, p. 359–362, 1981. Citation onpage 18.

ROGERS, C. L∞-algebras from multisymplectic geometry. Lett. Math. Phys, v. 100, p. 29–50,2012. Citation on page 51.

SMALE, S. Differentiable dynamical systems. Bull. Amer. Math. Soc., v. 73, p. 747–817, 1967.Citations on pages 11, 13, and 17.

STASHEFF, J. Citation on page 51.

TAYLOR, M. E. Noncommutative Microlocal Analysis, Part I. [S.l.]: AMS, 1984. (Memoirsof the American Mathematical Society). Citation on page 54.

TOMASSINI, A.; VEZZONI, L. Contact calabi-yau manifolds and special legendrian submani-folds. Osaka J. Math., v. 45, p. 127–147, 2008. Citations on pages 41 and 51.

VERJOVSKY, A. Codimension one anosov flows. Bol. Soc. Mat. Mexicana, v. 19, n. 2, p. 49 –77, 1974. Citation on page 18.

VITAGLIANO, L. L∞-algebras from multicontact geometry. Differential Geometry and itsApplications, v. 39, p. 147–165, 2015. Citations on pages 41 and 53.

UN

IVER

SID

AD

E D

E SÃ

O P

AULO

Inst

ituto

de

Ciên

cias

Mat

emát

icas

e d

e Co

mpu

taçã

o

· SERVIÇO DE PÓS-GRADUAÇÃO DO ICMC-USP Data de Depósito: Assinatura: ______________________...

Documents

Transcript of · SERVIÇO DE PÓS-GRADUAÇÃO DO ICMC-USP Data de Depósito: Assinatura: ______________________...