Cubics, curvature and asymptotics

Michael Pauley

This thesis is presented for the degree of

Doctor of Philosophy

of The University of Western Australia

Department of Mathematics & Statistics.

January 27, 2011

ii

Abstract

Geodesics are a generalisation of straight lines to Riemannian manifolds and other spaces

equipped with an affine connection. Interpolation and approximation problems motivate

analogous generalisations of cubic polynomials. There are several approaches.

Cubic polynomials in Euclidean space are critical points of the mean norm-squared

acceleration, motivating Riemannian cubics which are critical points of the mean norm-

squared covariant acceleration. Cubic polynomials are also curves of constant jerk, and this

motivates Jupp and Kent’s cubics or JK-cubics which are curves of covariantly constant

jerk.

This thesis contains results on Riemannian and JK-cubics. The primary concerns are

asymptotic properties of these curves for large time. There are results on integration of

the ordinary differential equations in special cases. JK-cubics and a special family of Rie-

mannian cubics known as null are studied in matrix groups. Asymptotics are described in

generic cases. We also consider these curves in Riemannian manifolds of strictly negative

curvature. In such manifolds we show the existence of an open family of Riemannian cubics

which cannot be extended for all time. We prove that all JK-cubics in Riemannian man-

ifolds of strictly negative curvature are asymptotically approximated by reparametrised

geodesics.

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iv

Acknowledgements

I thank Prof Lyle Noakes for supervising my PhD. It is not reasonable to try to reduce to

one paragraph a description of the incredible amount of time and patience Lyle has given

to me. I am indebted for the generosity Lyle has shown, with his wisdom, his time and

with research problems. I also appreciate the artistic freedom he has given me and his

kind encouragement.

Many others influenced this thesis through helpful comments and discussions, including

Shreya Bhattarai, Philip Schrader, Prof Andrew Bassom, Dr Tomasz Popiel, Asst/Prof

Mike Alder, Prof Lucho Stoyanov.

The following people (not listing those already mentioned) were useful in minimising

psychological attrition: my girlfriend, Joanne Chia, who is the kindest and most tolerant(!)

person I know; my wonderful mother (Janice P.), father (Gerald P.) and brother (Ben

P.); other housemates, past and present: Brian Corr, David Slacksmith, James Curry; the

residents of Currie Hall; Seyed Hassan Alavi, Neil Gillespie and the rest of the postgraduate

students and staff at the School of Mathematics and Statistics at UWA, who through their

kindness have made this time so enjoyable.

I am grateful for financial assistance from the Australian Postgraduate Award and the

Jean Rogerson Postgraduate Scholarship.

v

vi

Statement of candidate contribution

Chapter 7 is a paper submitted for publication, which I jointly authored, as primary

author, with my supervisor, Prof Lyle Noakes.

I had regular discussions on all aspects of all the other chapters, but not to the extent

of sharing co-authorship with Prof Noakes.

Michael Pauley (candidate):

Prof Lyle Noakes (supervisor and co-author):

vii

viii

Contents

Abstract iii

Acknowledgements v

Statement of candidate contribution vii

0 Introduction 1

1 Riemannian manifolds and Lie groups 5

1.1 Tensor fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Affine Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Semi-Riemannian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Riemannian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Sectional curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5.1 The hyperbolic plane . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.6 Jacobi fields. Manifolds of nonpositive and strictly positive curvature . . . . 11

1.7 Structure of Lie groups and Lie algebras . . . . . . . . . . . . . . . . . . . . 11

1.8 Affine connections on Lie groups . . . . . . . . . . . . . . . . . . . . . . . . 13

1.9 Bi-invariant metrics on Lie groups . . . . . . . . . . . . . . . . . . . . . . . 15

2 Literature review 19

2.1 Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.1 Interpolation problem . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.2 Smoothing problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2 Generalisations to curved spaces . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Riemannian cubics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.1 Lax constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4 Generalisations of Riemannian cubics . . . . . . . . . . . . . . . . . . . . . . 25

2.4.1 Riemannian cubics in tension . . . . . . . . . . . . . . . . . . . . . . 25

2.4.2 Higher order geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5 Jupp and Kent’s cubics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.6 Overview of results in this thesis . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Lax constraints in arbitrary Lie groups 29

ix

4 Jupp and Kent’s quadratics in Lie groups 33

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2 JK-quadratics in Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.3 SO(3) and SO(1, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.4 Null Lie quadratics and the Schrodinger equation . . . . . . . . . . . . . . . 37

4.5 Higher derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.6 SU(2), SO(1, 2,R), SO(3,R) and their Lie algebras . . . . . . . . . . . . . . 39

4.7 Euler spirals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.8 JK-quadratics in the sphere and hyperbolic plane . . . . . . . . . . . . . . . 41

4.9 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5 Jupp and Kent’s cubics in Lie groups 45

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.2 JK-Cubics in Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.2.1 (-)-connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.2.2 (0)-connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.3 Asymptotics of the linear system under generic conditions . . . . . . . . . . 49

5.4 (0)-connection: Conversion to two systems of linear ODEs . . . . . . . . . . 50

5.5 Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.6 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6 Lie algebras and asymptotic series for solutions of linear ordinary dif-

ferential equations 55

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.2 Application to null Riemannian cubics . . . . . . . . . . . . . . . . . . . . . 57

6.3 Application to JK-cubics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.4 Formal series and their exponentials . . . . . . . . . . . . . . . . . . . . . . 58

6.5 Formal solution: nilpotent Lie algebra . . . . . . . . . . . . . . . . . . . . . 64

6.6 Formal solution: arbitrary Lie algebra . . . . . . . . . . . . . . . . . . . . . 65

7 Cubics and negative curvature 67

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

7.2 Preliminary calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7.3 Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

7.4 A simplification in locally symmetric spaces . . . . . . . . . . . . . . . . . . 73

7.5 An example with exact F00 . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

7.6 Locally symmetric spaces with non-negative curvature . . . . . . . . . . . . 74

8 Geodesic curvature and asymptotic geodesics 77

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

8.2 Some lemmas in simply connected manifolds of non-positive curvature . . . 78

8.3 The theorem and its proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

8.4 Condition (8.2) is sufficient in the n-sphere . . . . . . . . . . . . . . . . . . 84

8.5 Application to JK-cubics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

x

9 Conclusions 87

Bibliography 89

Index 95

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Chapter 0

Introduction

In this thesis we study the asymptotics of two classes of curves which generalise cubics to

Riemannian manifolds and Lie groups.

Geodesics are a generalisation of straight lines to Riemannian manifolds. A first defi-

nition of a geodesic in a Riemannian manifold is “the shortest path between two points”.

There are a couple of reasons that this definition isn’t always appropriate. The first is

that it does not place any restrictions on the parametrisation of the curve. If a shortest

path is parametrised so that it has constant speed, then it is also a minimum of the mean

kinetic energy among curves with given endpoints. Thus, if we are only concerned with

geodesics of constant speed, a better definition might be “the path of minimal energy

between two points”. This still might not be what we want. “Critical points of energy

among curves with given endpoints” allows us to include non-minimal geodesics (e.g., for

two non-antipodal points in a sphere, the great circle containing them can be separated

into two arcs; one of these is minimal, the other is a critical point of energy but is not

minimal). It also allows us to speak of geodesics in a semi-Riemannian manifold. But in

defining geodesics as critial points of energy, we only allow ourselves to talk of geodesics

on a finite subinterval of R. Sometimes we would like our geodesics to be defined on the

whole real line (indefinitely extended). For example, an indefinitely extended geodesic in

a surface might describe the motion of a particle which is under no forces except for those

keeping it on the surface.

Instead of characterising geodesics by variational problems it is convenient to define

them as curves satisfying Euler-Lagrange equations. An affine connection on a manifold is

a method for differentiating tangent vector fields along curves. With an affine connection

we can differentiate the velocity to give what is called the covariant acceleration. Then

a geodesic is defined as a curve whose covariant acceleration is zero. In a Riemannian

manifold there is a suitable choice of affine connection – the Levi-Civita connection – for

which zero covariant acceleration is the Euler-Lagrange equation for energy. The Hopf-

Rinow Theorem generalises two of Euclid’s axioms. It says that in a path-connected,

complete (as a metric space) Riemannian manifold, geodesics are indefinitely extendible,

and there exists a (not necessarily unique) minimal geodesic between any two points.

Affine connection spaces are manifolds which have a covariant differentiation of tan-

gent vector fields, but it might not be derived from the Levi-Civita connection for a semi-

Riemannian metric. Matrix groups have an affine connection called the (0)-connection,

1

Figure 1: Left: two points p0, p1 in Euclidean space, and vectors v0, v1 at those points.

Right: the curve with minimal mean norm-squared acceleration, among all curves y

with y(0) = p0, y(1) = p1, and derivatives y(1)(0) = v0, y(1)(1) = v1.

Figure 2: Solutions, with identical initial conditions, of two fourth order differential

equations in the sphere. A JK-cubic (darker curve) and a Riemannian cubic (lighter

curve).

which has excellent properties, but it only arises from a semi-Riemannian metric under

narrow conditions on the structure of the group. We would like to treat matrix groups

and a generalisation, Lie groups, in this thesis. Defining geodesics as curves of zero co-

variant acceleration allows us to study geodesics in the more general situation of an affine

connection space.

We can study geodesics as solutions of an initial value, rather than boundary value,

problem. In a complete Riemannian manifold the study of long term dynamics of geodesics

has led to a large and rich theory (see for example [31, 62]).

Thinking of geodesics as a generalisation of straight lines motivates us to study general-

isations of other classes of curves to arbitrary Riemannian manifolds and affine connection

spaces. For example, elastic curves in Euclidean space are the critical points of mean

squared curvature, subject to boundary conditions. This can be generalised to a Rieman-

nian manifold using the geodesic curvature of a curve (see for example [39, 69]).

The curves studied in this thesis are two generalisations of cubics, formed by replac-

ing higher derivatives by covariant derivatives. Cubics in Euclidean space are curves

of constant jerk. On a finite interval, cubics are the critical points of the mean norm-

squared acceleration, among curves with specified endpoints and end velocities (see Fig-

ure 1). Riemannian cubics are critical points of mean norm-squared covariant accelera-

tion ([24, 38, 57]). Jupp and Kent’s cubics or JK-cubics are curves of covariantly con-

stant jerk ([38]). Examples of a JK-cubic and a Riemannian cubic are given in Figure 2.

Riemannian cubics and Jupp and Kent’s cubics are motivated by problems in computer

graphics, robotic control and statistics that need splines for interpolating or smoothing

2

data in curved spaces. Like geodesics, JK-cubics can be defined in any affine connection

space. By replacing the variational problem for Riemannian cubics with its Euler-Lagrange

equation, we can also study Riemannian cubics in any affine connection space. Even in

manifolds where geodesics can be solved in closed form, it is not necessarily possible to

solve for either kind of cubic. When exact solutions of ordinary differential equations are

not possible, we can approximate curves locally in time using, for example, Taylor series

(“local asymptotics”). As we increase the size of the time interval in which we want an ap-

proximation, we need more terms of a Taylor series, and the computational cost increases.

We would like some way to describe the nature of the solutions which can take over from

local methods once time gets sufficiently large. In other words, we would like to know the

asymptotics of solutions as t→ ±∞. An ideal asymptotic result would be: a solution of a

certain ordinary differential equation is approximated by a curve chosen from some exactly

solved or otherwise easily described class. We are motivated by [51, Theorem 7] which is

concerned with a special class of Riemannian cubics in Lie groups which are called null.

In the rotation group SO(3), for any null Riemannian cubic x, there is a geodesic γ such

that d(x(t), γ) converges to zero as t→∞. That is, x is asymptotically approximated by

a (reparametrised) geodesic.

In this thesis we prove results about the asymptotics of Riemannian Cubics and Jupp

and Kent’s cubics. We allow ourselves a couple of concessions. We do not attempt to

say which initial conditions lead to a given asymptotic solution. Quite often we will be

content with results that only hold for a generic family of solutions. By “generic” we

mean that there is a hypothesis (on, say, the initial conditions) which is true for an open

dense subset of the solutions. We visualise that there is only a thin slice of solutions for

which the hypothesis fails. In particular, results often require a certain polynomial to take

a nonzero value on the initial conditions. Sometimes we will be happy with asymptotics

in a special case. JK-cubics are simpler than Riemannian cubics and we expect stronger

results for them.

Having said all that about asymptotics, the early results (Chapters 3–4) are on in-

tegration, namely, obtaining more or less explicit solutions for Riemannian cubics and

JK-cubics. The differential equation for a Riemannian cubic in a Lie group is often stud-

ied by its Lie reduction, a related curve in the Lie algebra. Solution of a generic family of

Riemannian cubics by quadrature from their Lie reductions grew out of results for partic-

ular Lie groups to a general method for arbitrary semisimple Lie groups given in [54]. We

show how a modification of this method allows it to be applied to arbitrary Lie groups.

There is a special case of JK-cubics which correspond to parabolas. We study these

curves – JK-quadratics – in Lie groups and show how they are closely related to null Rie-

mannian cubics. In the 3-dimensional rotation group SO(3) and a related group SO(1, 2)

we show how JK-quadratics and null Riemannian cubics can be solved in terms of a quan-

tum harmonic oscillator , a special case of the Schrodinger equation in one dimension.

Results on the asymptotics of JK-cubics in matrix groups are proved in Chapters 5–6.

It is shown that JK-cubics in an n-dimensional matrix group can be solved in terms of

the solutions of a pair of n-dimensional linear systems of ordinary differential equations.

Asymptotic expansions can immediately be given for a generic family of JK-cubics in the

3

Figure 3: A surface of negative (sectional) curvature. Geodesics repel each other. (See

Section 1.5.)

general linear group GL(n). Motivated by this, we come up with a new way of writing

asymptotic expansions which works for a generic family of JK-cubics in an arbitrary matrix

group. This approach also gives asymptotics for generic null Riemannian cubics in matrix

groups.

In Chapters 7–8 we abandon the symmetry of Lie groups, and make some explorations

in Riemannian manifolds with strictly negative sectional curvature. This is a condition

on the curvature of the manifold which says that geodesics repel each other. (See figure

3.) The prime example of such a manifold is the hyperbolic plane. Since geodesics in

an arbitrary complete Riemannian manifold can be extended indefinitely, we would like

to know whether Riemannian cubics also can be extended indefinitely. Previously the

only result in non-flat manifolds was that Riemannian cubics are indefinitely extendible

in the rotation group SO(3) ([52]). We find that in manifolds of strictly negative sectional

curvature, Riemannian cubics are not always extendible to the whole real line – there

always exists an open family of Riemannian cubics whose velocities diverge to infinity in

finite time.

We prove that JK-cubics in manifolds of strictly negative curvature are always asymp-

totically approximated by reparametrised geodesics. In order to do this we generalise a

result of [23]. The (absolute) geodesic curvature k(t) of an arc-length parametrised curve

x is the absolute value of the (covariant) rate of change of its tangent vector. In the hyper-

bolic plane, if the total absolute curvature∫∞t0k(t)dt is finite, then x has an asymptotic

geodesic. We extend this to any manifold of strictly negative sectional curvature.

4

Chapter 1

Riemannian manifolds and Lie groups

In Sections 1.1–1.6 we summarise the necessary definitions and results from Riemannian

geometry. Much of this is based on [71], and some of it on [15, 16, 22, 49]. We take for

granted the concepts of smooth manifolds, smooth vector fields and their Lie brackets,

and sections of a vector bundle. In Sections 1.7–1.9 we cover the necessary background

on Lie groups and Lie algebras.

1.1 Tensor fields

The dual space of a vector space V is denoted V ∗. A tensor of type (a, b), or (a, b)-tensor,

on a finite dimensional vector space V is a multilinear map

S : V × · · · × V︸ ︷︷ ︸a times

×V ∗ × · · · × V ∗︸ ︷︷ ︸b times

→ R.

An element of V ∗ is, by definition, a (1, 0)-tensor. An element X of V can be identified

with the (0, 1)-tensor which maps any element θ of V ∗ to θ(X). More generally, an (a, 1)-

tensor T can be identified with the map which takes a vectors X1, · · · , Xa to the unique

vector Y which satisfies θ(Y ) = T (X1, · · · , Xa, θ) for any θ ∈ V ∗.Let π : E → M be a smooth vector bundle. Suppose S is a map assigning to each

point p ∈ M an (a, b) tensor on the fibre π−1(p). We say S is an (a, b)-smooth tensor

field (or just tensor field or even tensor) on M if, whenever X1, · · · , Xa are smooth vector

fields, and θ1, · · · , θb are smooth 1-forms, the function

S(X1, · · · , Xa, θ1, · · · , θb) : M → R

given by p 7→ S(X1(p), · · · , Xa(p), θ1(p), · · · , θb(p)) is smooth.

1.2 Affine Connections

An affine connection on a smooth manifold M is a map ∇ from the vector space of smooth

tangent vector fields to the vector space of (1, 1) tensor fields on the tangent bundle, with

the following properties (see, for example, [16, Chapter 4] or [49, Chapter 8]):

• ∇ is additive: for vector fields X1, X2, we have

∇(X1 +X2) = ∇(X1) +∇(X2); (1.1)

5

• ∇ satisfies the following product rule: for a function f : M → R and a vector field

X,

∇(fX) = (X)(df) + f∇(X). (1.2)

Applying the product rule when f is a constant k, we see that ∇(kX) = k∇(X). Together

with additivity this means that ∇ is linear. Since ∇(X) is a (1, 1)-tensor field, we can

identify its value ∇(X)p at any point p with a map TpM → TpM . Write ∇Y (X) for

the image of Y ∈ TpM under this map. It is called the covariant derivative of X in

the direction of Y at p. A manifold M with a covariant derivative ∇ is called an affine

connection space.

The torsion of an affine connection ∇ assigns the vector field

T (X,Y ) = ∇XY −∇YX − [X,Y ]

to two vector fields X,Y on M . The torsion T (X,Y )p at a point p depends only on the

values of X and Y at p, and therefore defines a (2, 1)-tensor field. If T (X,Y ) = 0 for all

vector fields X,Y we say that ∇ is torsion-free (or symmetric).

The Riemannian curvature tensor (or just curvature tensor) of ∇ is

R(X,Y )Z := ∇X∇Y Z −∇Y∇XZ −∇[X,Y ]Z.

The curvature (R(X,Y )Z)p at a point p also only depends on the values of X,Y, Z at p.

Thus R is a (3, 1)-tensor field.

Let ∇ be an affine connection on M and x : R → M be a smooth curve. Suppose

X : R→ TM is a smooth curve such that, for any t we have X(t) ∈ Tx(t)M . We call X a

smooth lifting of x to the tangent bundle, or a vector field defined along x.1 The derivative

x(1) is a lifting of x. There is a unique map

∇t : vector fields defined along x → vector fields defined along x

satisfying

∇t(X1 +X2) = ∇t(X1) +∇t(X2) for vector fields X1, X2 defined along x,

∇t(fX) =df

dtX + f∇tX for a vector field X and a function f : R→ R,

and with the following property: whenever X is a vector field on M , we have

∇t(X x)(t) = ∇x(1)(t)X for all t.

The operator ∇t is the covariant derivative along x. It is also written ∇d/dt or Ddt . Similar

definitions can be made for vector fields defined along maps Rn ⊇ U → M and their

partial derivatives. A vector field along a map x : (u1, · · · , un) 7→ x(u1, · · · , un) ∈ M is a

lifting X : U → TM of x. The covariant derivative ∇uiX at a point (u1, · · · , un) is equal

to the covariant derivative along the curve for which uj , (j 6= i) is fixed.

1If we took the phrase “vector field defined along x” literally, self-intersections of the curve x would be

a problem: if t1 6= t2 such that x(t1) = x(t2) then we would also require X(t1) = X(t2), but we don’t.

6

If X is a vector field on M such that, for every vector field Y we have ∇YX = 0 then

we say that X is a parallel vector field.2 Let x : R→M be a smooth curve, and let X be

a vector field defined along x such that ∇tX = 0. Then we say that X is parallel along

the curve x. For fixed t0 and any A ∈ Tx(t0)M , there is exactly one vector field X defined

along x such that X is parallel along x and X(t0) = A. The value of X at some other

time t is called the parallel translation of A along x from time t0 to time t.

Let I be a (finite or infinite) subinterval of R and let γ : I → M be a curve in an

affine connection space M . The vector field ∇tγ(1) is called the covariant acceleration of

γ and if ∇tγ(1)(t) = 0 for all t, we call γ a geodesic. In other words, γ is a geodesic if

its derivative γ(1) is a parallel vector field along γ. Whenever γ(t) is a geodesic, so is the

curve t 7→ γ(λt) for any real number λ.

Let p ∈ M and let A be a tangent vector at p. As explained in, for example, [70,

Section 2.6], there is a geodesic γ defined in a neighbourhood of 0 with γ(0) = p and

γ(1)(0) = A. Any two geodesics with these initial conditions agree (on the intersection of

their domains). Let D be the set of of all A ∈ TpM such that there is a geodesic γ defined

on [0, 1] with the initial conditions γ(0) = p and γ(1)(0) = A. Define

expp(A) := γ(1).

This is the exponential mapping from D to M .

1.3 Semi-Riemannian manifolds

Let V be a vector space. A (2, 0)-tensor (or bilinear form) is often written 〈·, ·〉. The

bilinear form 〈·, ·〉 is called symmetric if 〈x, y〉 = 〈y, x〉 for all x, y ∈ V . We say 〈·, ·〉 is

degenerate if there is some nonzero x ∈ V such that 〈x, y〉 = 0 for every y ∈ V , otherwise

we say 〈·, ·〉 is nondegenerate. A (pseudo- or) semi-Riemannian metric on a manifold M

is a smooth symmetric (2, 0)-tensor field which is nondegenerate at each point of M . A

smooth manifold with a semi-Riemannian metric is called a semi-Riemannian manifold .

A connection ∇ on M is called metric or metric compatible if

X〈Y,Z〉 = 〈∇XY, Z〉+ 〈Y,∇XZ〉

for any smooth vector fields X,Y, Z. If ∇ is a metric connection then for vector fields Y, Z

defined along a curve x we have ddt〈Y,Z〉 = 〈∇tY, Z〉+ 〈Y,∇tZ〉.

Theorem 1.1 (Fundamental Theorem of Riemannian Geometry). There is exactly one

torsion-free metric connection on a given semi-Riemannian manifold. It is called the

Levi-Civita connection.

Let M be a semi-Riemannian manifold. Suppose that N is a submanifold of M . At

any point p ∈ N the vector space TpN is a subspace of TpM so N has a symmetric (2, 0)-

tensor defined by the restriction of 〈·, ·〉. It is possible that this tensor is degenerate. If it

is nowhere degenerate, N is a semi-Riemannian manifold with this metric. (See e.g. [60,

2The zero vector field on any affine connection space is parallel, but many affine connection spaces have

no nonzero parallel vector fields.

7

Chapters 3-4]). Then TpM can be decomposed into TpN and its orthogonal complement

(TpN)⊥. This defines a projection map projTpN : TpM → TpN . If D is the Levi-Civita

connection on N then for vector fields X,Y defined on a neighbourhood of N ,

(DXY )p = projTpN (∇XY )p, (1.3)

and for a curve x : R→ N and a vector field X defined along x,

(DtX)(t0) = projx(t0)(∇tX)(t0). (1.4)

1.4 Riemannian manifolds

A Riemannian metric on a smooth manifold M is a symmetric (2, 0)-tensor field 〈·, ·〉which is positive definite at every point, i.e. for any vector X at a point p, the number

〈X,X〉 is non-negative, and 〈X,X〉 = 0 ⇐⇒ X = 0. A manifold with a Riemannian

metric is called a Riemannian manifold . The condition of positive definiteness makes 〈·, ·〉nondegenerate, so any Riemannian manifold is a semi-Riemannian manifold.

A Riemannian metric defines a norm ‖A‖ :=√〈A,A〉 for any vector A in the tangent

space at any point. Let M be a Riemannian manifold and let x : [0, 1]→M be a smooth

curve. The length of x is defined as

L(x) =

∫ 1

0‖x(1)(t)‖dt. (1.5)

If x1, x2 : [0, 1]→M are curves such that x2 is a reparametrisation of x1 (x2(t) = x1(φ(t))

for some smooth increasing bijection φ : [0, 1] → [0, 1]), then x1 and x2 have the same

length. Suppose M is path-connected. If p, q ∈M we define

d(p, q) := inf L(x) : x is a smooth curve with x(0) = p, x(1) = q (1.6)

In this way a Riemannian metric gives us a metric (in the usual sense). It is possible

that the distance d(p, q) may be realised by some curve x. In this case the distance

is also realised by any reparametrisation of x, and in particular it is realised by the

reparametrisation with constant speed d(p, q).

Given a curve x : [0, 1] → M with x(0) = p, x(1) = q, extend x to a variation or

homotopy through curves satisfying the boundary conditions: now x is a smooth map from

[1, 0]× (−ε, ε)→M such that x(t, 0) agrees with the original curve x(t), while x(0, u) = p

and x(1, u) = q for all u. If x(·, 0) minimises the integral (1.5) then it minimises (1.5) over

all the curves x(·, u) for fixed u. Thus dLdu |u=0 = 0. Instead of looking just at minimisers

of L, we consider critical points, i.e., curves such that dLdu |u=0 = 0 for all variations of x.

Critical points of L with constant speed turn out to also be critical points of the energy

E =

∫ 1

0〈x(1)(t), x(1)(t)〉dt. (1.7)

Write xu, xt for the partial derivatives of x. Let ∇ be the Levi-Civita connection. Com-

puting dEdu using integration by parts and the boundary conditions,

dE

du

∣∣∣∣u=0

= 2

∫ 1

0〈∇uxt, xt〉dt

∣∣∣∣u=0

= −2

∫ 1

0〈xu,∇txt〉dt

∣∣∣∣u=0

.

8

So, using the fundamental lemma of the calculus of variations, we have ∇tx(1) = 0; thus

x is a geodesic. We say that x is a minimal geodesic if the length of x is d(p, q).

Let us say that a geodesic x : I →M is extendible to R if there is a geodesic x1 : R→M

with x1|I ≡ x. We say that M is geodesically complete if every geodesic is extendible to

R. In other words, for every p, the exponential mapping expp is defined on all of TpM .

Theorem 1.2 (Hopf-Rinow). A path-connected Riemannian manifold M is complete as

a metric space if and only if it is geodesically complete. If M is complete, then any points

p, q are connected by a minimal geodesic.

1.5 Sectional curvature

For a vector space V , write Λ2V for the vector space of all (0, 2)-tensors P such that

P (θ1, θ2) + P (θ2, θ1) = 0 for all θ1, θ2 ∈ V ∗. For X1, X2 ∈ V the wedge product X1 ∧X2

defined by (X1 ∧X2)(θ1, θ2) = θ1(X1)θ2(X2)− θ2(X1)θ1(X2) is an element of Λ2(V ), and

Λ2(V ) is spanned by elements of this form. (See for example [2, Section 2.3.5].) If an inner

product 〈·, ·〉 on V is given, then it defines an inner product on Λ2V by

〈X ∧ Y, Z ∧W 〉 = 〈X,Z〉〈Y,W 〉 − 〈X,W 〉〈Y,Z〉,

and extending by linearity. The inner product is positive definite by the Cauchy-Schwarz

inequality. As usual the norm ‖P‖ is defined as√〈P, P 〉. In Euclidean space, ‖X ∧ Y ‖ is

the area of the parallelogram with sides X and Y .

Let M be a Riemannian manifold, and let ∇ be its Levi-Civita connection. The

Riemannian curvature tensor R satisfies the following identities, for any p ∈ M and for

any X,Y, Z,W ∈ TpM (See e.g. [71, Chapter 15]).

R(X,Y )Z +R(Y,X)Z = 0; (1.8)

R(X,Y )Z +R(Y,Z)X +R(Z,X)Y = 0; (1.9)

〈R(X,Y )Z,W 〉+ 〈R(X,Y )W,Z〉 = 0; (1.10)

〈R(X,Y )Z,W 〉 = 〈R(W,Z)Y,X〉. (1.11)

Suppose X,Y,X ′, Y ′ are given such that span(X ′, Y ′) = span(X,Y ). Then, writing X ′ =

aX + bY and Y ′ = cX + dY , we see from (1.8) and (1.10) that

〈R(X ′, Y ′)Y ′, X ′〉 = (ad− bc)2〈R(X,Y )Y,X〉.

But

‖X ′ ∧ Y ′‖2 = (ad− bc)2‖X ∧ Y ‖2.

This means that for a given 2-dimensional subspace σ of TpM , the ratio

K(σ) = K(X,Y ) =〈R(X,Y )Y,X〉‖X ∧ Y ‖2

(1.12)

is independent of the choice of X,Y spanning σ. the number K(σ) is called the sectional

curvature of M in the plane σ at the point p.

9

1.5.1 The hyperbolic plane

Define on R3 the bilinear form

⟨x1x2x3

,

y1y2y3

⟩ := −x1y1 + x2y2 + x3y3. (1.13)

The bilinear form 〈·, ·〉 is symmetric and nondegenerate. The tangent space of R3 at any

point is R3 itself, and 〈·, ·〉 defines a semi-Riemannian metric on R3. The usual directional

derivative is torsion free and metric compatible, so it is the Levi-Civita connection for this

metric. The set x : 〈x, x〉 = −1 is a two-sheeted hyperboloid: it has two path-connected

components. In one component all points have x1 ≥ 1, and in the other all points have

x1 ≤ −1. The component with all x1 ≥ 1 is called the hyperbolic plane H2. If p ∈ H2

and X is a nonzero tangent vector at p then 〈p,X〉 = 0 and (1.13) implies 〈X,X〉 > 0.

Thus H2 is actually a Riemannian manifold with the restriction of 〈·, ·〉. We compute the

Riemannian curvature tensor. Let X,Y, Z be vectors at a point p ∈ H2. Choose a map

(u, v) 7→ x(u, v) from R2 to H2 such that x(0) = p, xu(0) = X,xv(0) = Y . Extend Z to a

vector field defined on an open neighbourhood of p in H2.

Suppose W is any vector field defined along x. Then 〈W,x〉 ≡ 0 since W is a tangent

vector field. Since the covariant derivative ∇ is found by projecting the usual derivative

onto tangent spaces, we have for any vector field W defined along x (see Equation (1.4)):

∇vW = Wv −〈Wv, x〉〈x, x〉

x = Wv − 〈W,xv〉x.

Similarly, ∇uW = Wu − 〈W,xu〉x. So

∇u∇vZ = Zvu − 〈Zu, xv〉x− 〈Z, xvu〉x− 〈Z, xv〉xu − 〈Zv − 〈Z, xv〉x, xu〉x.

We get a similar expression for ∇v∇uZ. Combining, and using [xu, xv] ≡ 0,

R(xu, xv)Z = −〈Z, xv〉xu + 〈Z, xu〉xv.

Setting (u, v) = (0, 0) gives R(X,Y )Z = −〈Z, Y 〉X + 〈Z,X〉Y. Therefore the sectional

curvature at any point is

K(X,Y ) =〈R(X,Y )Y,X〉

〈X,X〉〈Y, Y 〉 − 〈X,Y 〉2= −1.

Curves in the hyperbolic plane are sometimes drawn using a stereographic projection,

defined in [37, Section 1.6] as follows. Let u =

−1

0

0

. For a point p ∈ H2 connect u to

p via a line segment in R3 and let φ(p) be the intersection of this line segment with the

plane orthogonal to u (in this case the plane x1 = 0). The stereographic projection maps

H2 to the disc x ∈ R2 : ‖x‖ < 1. (See figure 1.1.) This representation of the hyperbolic

plane is called the Poincare disc. Later, in Figure 2.1, we show a JK-cubic in the Poincare

disc.

10

p

φ(p)

Figure 1.1: Stereographic projection of the hyperbolic plane.

1.6 Jacobi fields. Manifolds of nonpositive and strictly positive curvature

A map x : [0, 1]× (−ε, ε)→ M is called a variation through geodesics, if for each fixed u,

the map t→ x(t, u) is a geodesic. Write xt and xu for the partial derivatives in the t and

u directions. Then since ∇ is torsion-free,

∇2txu = ∇t∇uxt = ∇u∇txt +R(xt, xu)xt +∇[xt,xu]xt.

But ∇txt ≡ 0 and [xt, xu] ≡ 0, so using (1.8), ∇2txu = −R(xu, xt)xt. A vector field

J(t) along a geodesic γ(t) is called a Jacobi field if it satisfies the second order ordinary

differential equation,

∇2tJ = −R(J, γt)γt.

Thus, if x(t, u) is a variation through geodesics, then for any u the vector field xu is a

Jacobi field along the geodesic γ(t) = x(t, u). Jacobi fields are the essential ingredient in

the proof of the following theorem (and many others).

Let M,N be n-dimensional connected manifolds. A smooth map f : M → N is called

a covering mapping if there is an open cover Uα of N such that, for each Uα in the

cover, the preimage f−1(Uα) is a disjoint union of open sets Vα,β such that for each β

the restriction of f to Vα,β is a diffeomorphism Vα,β → Uα.

Theorem 1.3 (Cartan-Hadamard). Let M be an n-dimensional complete connected Rie-

mannian manifold with K(σ) ≤ 0 for every plane σ at every point. Then, for any p,

the exponential mapping expp : TpM → M is a covering mapping. In particular, if M is

simply connected then M is diffeomorphic to Rn.

1.7 Structure of Lie groups and Lie algebras

For the necessary theory (summarised here) of Lie groups and Lie algebras see [34, 41, 75,

83]. A (real or complex) Lie algebra is a vector space g (over R or C) equipped with a Lie

bracket which is a bilinear map [·, ·] : g× g→ g satisfying

[X,Y ] + [Y,X] = 0 (anticommutativity),

[X, [Y,Z]] + [Y, [Z,X]] + [Z, [X,Y ]] = 0 (Jacobi identity),

11

for all X,Y, Z ∈ g. The set of vector fields on any manifold, with the usual Lie bracket, is a

(real) Lie algebra (infinite dimensional except on zero-dimensional manifolds). If g1, g2 are

Lie algebras, a homomorphism is a map φ : g1 → g2 which is linear and satisfies φ[X,Y ] =

[φX, φY ]. A homomorphism φ is an isomorphism if φ is a vector space isomorphism. A Lie

algebra is abelian if [X,Y ] = [Y,X] for all X,Y . By anticommutativity this is equivalent

to [·, ·] ≡ 0.

For subalgebras h1, h2 of a Lie algebra g we write [h1, h2] for the vector space spanned

by all elements of the form [X1, X2] for X1 ∈ h1, X2 ∈ h2. We say that a subalgebra h of

g is an ideal of g if [g, h] ⊆ h. A Lie algebra g is simple if its dimension is at least 2 and

0, g are the only ideals of g.

For X ∈ g define ad(X) : g → g by ad(X)Y = [X,Y ]. The linear map ad maps g

to the algebra End(g) of all vector space endomorphisms of g, and is called the adjoint

representation of g. Maps ad(X) are derivations of the Lie algebra, namely

ad(X)[Y, Z] = [ad(X)Y, Z] + [Y, ad(X)Z],

which follows by rewriting the Jacobi identity.

The Killing form (or Cartan-Killing form) is a symmetric bilinear form 〈·, ·〉 on g

defined by

〈X,Y 〉 = Tr((adX)(adY )).

If 〈·, ·〉 is nondegenerate we say that g is semisimple. A semisimple Lie algebra is a direct

sum of simple Lie algebras.

On the other hand, a Lie algebra g is nilpotent if ad(X) is a nilpotent endomorphism

of g for every X. If g is nilpotent and has dimension m then [83, Corollary 3.5.6] there

are ideals gi, i = 0, · · ·m of g such that g0 = g ⊇ g1 ⊇ · · · ⊃ hm = 0, the dimension of gi

is m− i, and for i = 0, · · · ,m− 1 we have [g, hi] ⊆ hi+1.

Let g be an arbitrary finite dimensional Lie algebra. A Cartan subalgebra (CSA) is a

subalgebra h of g such that (i) h is nilpotent; (ii) h is its own normalizer in g, i.e. the set

X ∈ g : [X,Y ] ∈ h for all Y ∈ h

is equal to h. Basic facts about Cartan subalgebras are given in [83, Lemma 4.1.1 and

Theorem 4.1.2]. A CSA h of g is maximal nilpotent, i.e. there is no nilpotent n with

h $ n $ g. For X ∈ h, since ad(X)Y ∈ h for all Y ∈ h, it makes sense to consider

the induced map ad(X)g/h on the quotient space g/h. The determinant det(ad(X)g/h) is

a polynomial function on h which is not identically zero. The rank of g is the smallest

integer r such that the coefficient of λr in the characteristic polynomial det(λ − adX) is

nonzero for some X ∈ g. The coefficient of λr is some polynomial function η(X). An

element X of g is called regular if η(X) is nonzero. If X is regular then

hX = Y ∈ g : ad(X)s(Y ) = 0 for some positive integer s

is a CSA, and for Y ∈ hX we have η(Y ) = det(ad(Y )g/hX ). The dimension of any

CSA is equal to the rank of g. If g is semisimple then [83, Theorem 4.1.5] a CSA h is

12

maximal abelian and the restriction of the Killing form to h is nondegenerate. CSAs are

a fundamental tool in the classification of semisimple Lie algebras.

A Lie group is a smooth manifold G with a group structure such that the multiplication

map G × G → G and the inversion map G → G are smooth. (Thanks to a Theorem of

Cartan, we can assume G is a real analytic manifold, and multiplication and inversion are

real analytic [71, Section 7.8].) For our purposes all Lie groups are finite dimensional.

Let G be a Lie group. Write 1 for the identity of G. For an element g of G define the

maps

Lg : G→ G Rg : G→ G

h 7→ gh h 7→ hg.

The derivative (dLg)h of Lg at the point h ∈ G maps the tangent space ThG to the tangent

space TghG. For g1, g2 ∈ G, the derivatives dLg1 and dRg2 commute in the sense that, for

X in the tangent space ThG at some point h ∈ G,

(dLg1)hg2 (dRg2)hX = (dRg2)g1h (dLg1)hX.

if f(t), g(t) are curves in G then

d

dt(f(t)g(t)) = (dLf(t))g(t)g

(1)(t) + (dRg(t))f(t)f(1)(t), (1.14)

which follows from e.g. [71, Equation 8.13]. As a consequence of (1.14),

d

dt

(g(t)−1

)= −(dLg(t)−1)1(dRg(t)−1)g(t)g

(1)(t). (1.15)

A vector field X on G is called left invariant if Xgh = (dLg)Xh. The Lie bracket [X,Y ]

of two left invariant vector fields is also left invariant. Thus the left invariant vector fields

form a finite dimensional subalgebra of the Lie algebra of all vector fields. This subalgebra

is called the Lie algebra g of G. Since a left invariant vector field is determined by its

value at the identity, we can identify g with the tangent space T1G.

A special case of a Lie group is when G is a real matrix group, namely a Lie group

consisting of real matrices with the usual multiplication. Then g is a matrix Lie algebra,

and the Lie bracket of left-invariant vector fields agrees with the commutator [X,Y ] =

XY − Y X of matrices in the tangent space at the identity. (Similar definitions can be

made for complex matrix groups.) The derivatives dLg and dRg are just given by X → gX

and X → Xg respectively, and Equations (1.14) and (1.15) reduce to the familiar

d

dt(f(t)g(t)) = f(t)g(1)(t) + f (1)(t)g(t), and

d

dt

(g(t)−1

)= −g(t)−1g(1)(t)g(t)−1.

1.8 Affine connections on Lie groups

For the theory of left-invariant connections on Lie groups, see [71, Chapter 6] or [32,

Section II.1.3].

There is an exponential mapping exp : g → G as follows. Let A ∈ g. Treat A as a

left invariant vector field on G. Let x(t) be the integral curve of A satisfying x(0) = 1.

13

The curve x exists for all time and is actually a group homomorphism from R to G. Let

exp(A) := x(1).

For g ∈ G define Ad(g) : g → g by Ad(g)X = (dLg)g−1 (dRg−1)1X. The map

Ad : G→ GL(g) is a smooth group representation. It is called the adjoint representation

of G. Every Ad(g) is an automorphism of g. Treating g as the tangent space at the

identity, we have

(d Ad)1 = ad, (1.16)

and, for any X ∈ g,

Ad(exp(X)) = exp(adX) (1.17)

Considering the curve t 7→ exp(tX) ∈ G, we have

d

dt

∣∣∣∣t=0

Ad(exp(tX))Y = ad(X)Y.

An affine connection ∇ on G is called left invariant if, whenever X and Y are left

invariant vector fields, the vector field ∇XY is also left invariant. Thus ∇ defines a

bilinear operator α : g× g→ g. In fact, every such bilinear map α : g× g→ g determines

a left invariant affine connection. The torsion of the connection can be computed on

tangent vectors X,Y at a point g: the torsion T (X,Y )g depends on X and Y only at g.

We can extend X,Y to left-invariant vector fields on G. Treating X,Y as elements of g,

compute

T (X,Y ) = ∇XY −∇YX − [X,Y ] = α(X,Y )− α(Y,X)− [X,Y ], (1.18)

and evaluate the resulting left-invariant vector field at g.

Strangely, the affine connection space definition of the exponential map at the identity

(on page 7) does not have to be the same as the Lie group definition of the exponential

map. They agree when

α(X,X) = 0 ∀X ∈ g. (1.19)

From (1.18), (1.19) we see that there is a unique torsion-free left-invariant affine connec-

tion ∇ for which the two definitions of exp agree. This connection is significant enough to

have three names: the canonical connection, the mean connection and the (0)-connection.

The map α satisfies

α(X,Y ) =1

2[X,Y ].

Let us compute the curvature of the (0)-connection, for vectors X,Y, Z at any point

g: extend X,Y, Z to left invariant vector fields on G. Then

R(X,Y )Z = ∇X∇Y Z −∇Y∇XZ −∇[X,Y ]Z =1

4([X, [Y,Z]]− [Y, [X,Z]])− 1

2[[X,Y ], Z]

= −1

4[[X,Y ], Z], (1.20)

(using the Jacobi identity) and evaluate at g.

14

Since Lg is a diffeomorphism G → G, a vector field X can be pushed forward to

a vector field Lg∗X defined by (Lg∗X)gh := LgXh. We say that Lg is an affinity if

∇Lg∗XLg∗Y = Lg∗(∇XY ). It turns out that the connection ∇ is left invariant if and only

if all the left multiplications Lg are affinities. If ∇ is left invariant, and letting α be the

map defined above, we can compute that right multiplication by an element g is an affinity

if Ad(g)α(X,Y ) = α(Ad(g)X,Ad(g)Y ) for all X,Y ∈ g. Thus a left-invariant affine

connection ∇ is bi-invariant (all left and right multiplications are affinities; equivalently,

for all left (resp. right) invariant X,Y , the covariant derivative ∇XY is left (resp. right)

invariant) when

Ad(g)α(X,Y ) = α(Ad(g)X,Ad(g)Y ) (1.21)

for all X,Y ∈ g and for all g ∈ G.

Because Ad(g) is an automorphism of the Lie algebra g, one easy way to make (1.21)

true is by letting α be a multiple of the bracket, α(X,Y ) = λ[X,Y ] for some constant

λ. The (0)-connection is of this form. Two other useful connections are the left or (-)-

connection, defined by

α(X,Y ) = 0

for which all left invariant vector fields are parallel; and the right or (+)-connection,

defined by

α(X,Y ) = [X,Y ]

for which all right invariant vector fields are parallel.

Let G be a group with a left invariant affine connection ∇ defined by some bilinear

map α : g × g → g. Let x : R → G be a smooth curve. For any vector field X defined

along x, let

X(t) = (dLx−1(t))x(t)X(t). (1.22)

Now X is a curve in the Lie algebra g. In particular let V (t) = x(1)(t); In [51], V is called

the Lie reduction of x. Let X be any vector field along x. Then a calculation shows that

∇tX(t) = X(1)(t) + α(V (t), X(t)). (1.23)

For example, a geodesic x satisfies ∇tx(1)(t) = 0. Applying (1.23) gives the equation

V (1)(t) + α(V (t), V (t)) = 0. But x is an integral curve of a left-invariant vector field only

if V (1)(t) = 0 for all t. This explains the claim above that the two definitions of the

exponential map agree only when α(X,X) = 0 ∀X ∈ g.

1.9 Bi-invariant metrics on Lie groups

We say that a semi-Riemannian metric 〈·, ·〉 on a Lie group G is left invariant if, for any

g, h ∈ G, and for any X,Y ∈ ThG, we have

〈(dLg)hX, (dLg)hY 〉 = 〈X,Y 〉.

15

We define a right-invariant semi-Riemannian metric similarly, and say that a metric is

bi-invariant if it is both left and right invariant.

Suppose 〈·, ·〉 is a bi-invariant metric, and X,Y ∈ g. Treating g as the tangent space

at the identity, 〈·, ·〉 becomes a nondegenerate bilinear form on g. For g ∈ G, since (dLg)

and (dRg−1) preserve the metric, we have

〈Ad(g)X,Ad(g)Y 〉 = 〈X,Y 〉. (1.24)

Substitute g = exp(tA) for some A ∈ g, and differentiate (1.24) at t = 0:

d

dt

∣∣∣∣t=0

〈Ad(exp(tA))X,Ad(exp(tA))Y 〉 = 0

so that

〈(adA)X,Y 〉+ 〈X, (adA)Y 〉 = 0. (1.25)

Equation (1.25) is called the ad-invariance of the metric 〈·, ·〉. Any ad-invariant non-

degenerate bilinear form on g determines a bi-invariant semi-Riemannian metric on G.

The Levi-Civita connection for a bi-invariant semi-Riemannian metric is bi-invariant and

torsion-free, so it is the (0)-connection, i.e. ∇XY = 12 [X,Y ] for left invariant X,Y .

The Killing form is ad-invariant but is sometimes degenerate. If g is semisimple, the

Killing form is nondegenerate and defines a semi-Riemannian metric on G. If the Killing

form is negative definite then a negative multiple gives a bi-invariant Riemannian metric

on G. In this case, G is compact [49, Section 21].

As an example, consider the rotation group SO(3) of all 3×3 matrices A of determinant

1, satisfying AAT = 1. The Lie algebra of SO(3) is so(3), consisting of all 3× 3 matrices

Ω with Ω + ΩT = 0, in other words, matrices of the form

Ω =

0 −c b

c 0 −a−b a 0

,

for real a, b, c. The linear isomorphism

φ : so(3)→ R3

Ω 7→

abc

transforms the Lie bracket (the commutator) of elements of so(3) into the cross product

of elements of R3. We can use this map to write ad(Ω) as a 3× 3 matrix. A computation

shows that this 3× 3 matrix is equal to Ω itself. We compute the Killing form:

⟨ 0 −c1 b1

c1 0 −a1−b1 a1 0

,

0 −c2 b2

c2 0 −a2−b2 a2 0

⟩ = −2(a1a2 + b1b2 + c1c2)

which is negative definite. Therefore the negative of the Killing form gives a Riemannian

metric on SO(3).

16

The 3-sphere S3 ⊆ R4 can be identified with the special unitary group in 2 dimensions:

this is the (real Lie) group of 2×2 complex matrices X satisfying XXT = 1 and det(X) =

1. Equivalently it is the group of unit quaternions. The Lie algebra of the 3-sphere

is isomorphic to so(3), so that SU(2) has a bi-invariant Riemannian metric. There is

a smooth group homomorphism S3 → SO(3) whose kernel is ±1. (See for example [81,

Chapters 1-2].) This group homomorphism is a local isometry: it preserves the Riemannian

metric.

17

18

Chapter 2

Literature review

2.1 Splines

2.1.1 Interpolation problem

Suppose a sequence of times t1 < · · · < tk ∈ R is given along with a sequence of points

x1, · · · , xk in n-dimensional Euclidean space En, which is Rn equipped with the Euclidean

inner product. We may desire a curve x which passes through each point xi at time ti

and has a reasonable level of smoothness. (Here, by smoothness, we just mean that the

curve visually does not appear too rough, for example, its second derivative is not too

large. Elsewhere in the present thesis “smooth” means C∞.) The theory of polynomial

splines [1, 20, 76] suggests choosing a curve whose restriction to each interval [ti, ti+1) is a

polynomial (i.e. each coordinate function t→ xi(t) is a polynomial on [ti, ti+1)). Allowing

higher degree polynomials gives extra degrees of freedom which can be used to make the

curve as many times differentiable as desired, and any remaining degrees of freedom can

be chosen to give the curve other desirable properties.

One could, in fact, make the whole curve a polynomial of degree at most k − 1. The

set of such polynomials forms a kn dimensional vector space, and the conditions x(ti) = xi

place kn affine equations on this space. It can be checked that the fact that t1 < · · · < tn

implies that the kn affine equations are consistent, so there is a unique solution. This

interpolant is called the Lagrange interpolating polynomial, and methods exist to compute

it efficiently. [20, Chapter II] points out some limitations of the Lagrange interpolating

polynomial. In particular if the data are badly behaved in just a small region, then the

polynomial will perform poorly everywhere. Using piecewise polynomials helps to avoid

this.

The simplest piecewise polynomial is piecewise degree 1, namely a concatenation of

line segments. Requiring that each line segment has endpoints xi and xi+1 determines

the curve uniquely. The resulting curve is the shortest path which passes through all the

points xi in order. The situation at hand may require a greater degree of smoothness, so

we consider curves which are piecewise quadratic and curves which are piecewise cubic.

In [20, Chapter VI] it is shown that in the case of parabolic splines, it is useful to

make the breakpoints different from the interpolation times, i.e. to use a curve which is

quadratic on intervals [ξi, ξi+1), i = 1, · · · , ik, where ξ1 ≤ t1 < ξ2 < t2 < · · · < tn−1 < ξn <

19

tn ≤ ξn+1. Then the polynomial pieces can be chosen so that the curve is differentiable

on [ξ1, ξn+1) and enjoys good approximation properties.

Piecewise cubics can be made twice differentiable on [t1, tn]. Such curves are what de

Boor labels cubic splines [20, Chapter IV]. Cubic splines have two extra degrees of freedom

which are often chosen by the endpoints. Natural cubic splines are cubic splines x with

x(2)(t1) = x(2)(tn) = 0. They minimise∫ tn

t1

〈x(2)(t), x(2)(t)〉dt (2.1)

among all differentiable curves satisfying x(ti) = xi for each i. The integral (2.1) is a suit-

able measure of the roughness of a curve, so minimising (2.1) seems to be an appropriate

way to optimise smoothness. (However de Boor points out that when using cubic splines

to approximate functions, other choices for end point conditions can be more effective.)

2.1.2 Smoothing problem

As in the interpolation problem, times t1 < · · · < tk ∈ R and points x1, · · · , xk ∈ En are

given, as well as a sequence w1, · · · , wk of weights for the data. This time a curve is desired

which passes near the point xi at time ti but not necessarily through it. In [20, Chapter

XIV] the smoothing spline of Schoenberg and Reinsch (for n = 1) is explained, and in [38,

Section 4.1] a higher dimensional version is given: it is the curve x which minimizes, for a

parameter p ∈ [0, 1],

p

k∑i=1

⟨xi − x(ti)

wi,xi − x(ti)

wi

⟩+ (1− p)

∫ tk

t1

〈x(m)(t), x(m)(t)〉dt. (2.2)

If the data xi come from an experiment, the weights wi may be estimates of the standard

error. Critical points of (2.2) are polynomial splines with breakpoints ti. They satisfy the

endpoint conditions x(j)(t1) = 0 = x(j)(tk) for j = m, · · · , 2m− 2. The number p specifies

a compromise between accuracy and smoothness. Choosing p near 1 encourages the curve

to travel close to the data; p near 0 encourages the curve to have small x(m).

2.2 Generalisations to curved spaces

For many practical interpolation and smoothing problems the data x1, · · · , xk come from

a curved space and the interpolating curve should stay in this space. For example, robotic

control [57], or animation [24], may require a curve in the rotation group SO(3). Directional

statistics, e.g. the statistics of magnetic polar wander, may require a smoothing spline in

the 2-sphere [38]. These spaces are not vector spaces, so polynomials (in the usual sense)

do not exist.

We interpret “curved space” to mean an affine connection space. In particular we are

interested in Riemannian manifolds, with the Levi-Civita connection, and Lie groups, with

the (0)-connection.

A first attempt at a solution may be to translate the data into coordinate charts and

interpolate there using polynomial splines. A pitfall of this approach is explained in [50].

In areas where the charts are close to local isometries this approach works fine, but when

20

the charts distort the space too much, the interpolant curves become very awkward. For

example, when the stereographic projection from the north pole is used as a chart for

the 2-sphere, the interpolant curves will favour the south pole even when the data points

x1, · · · , xk lie near the north pole.

Control points in the n-sphere can be treated as points in En+1. A second attempt

may be to use a polynomial interpolant curve in En+1 and normalise, pointwise, to get a

curve in the sphere. Besides the fact that this method does not generalise well to other

curved spaces (e.g. SO(3)), it still suffers drawbacks in the sphere [55]. Control points

that are too far apart lead to interpolant curves with wildly varying speed. The problem

is that the normalisation map En+1 → Sn distorts the metric too heavily, just like charts.

An appropriate general definition of interpolation curves in curved spaces should take

account of the geometry of the situation. A helpful guideline is that we prefer a defini-

tion that is intrinsic to the manifold in question, depending only on the metric or the

connection, and not on a choice of charts, or on an embedding in Euclidean space. We

expect also a sensible definition to specialise to a polynomial spline when the manifold is

Euclidean space with the usual metric. This has led to several generalisations of cubics

and other polynomials to Riemannian manifolds and affine connection spaces.

Cubic polynomials are critical points of the mean norm-squared acceleration. This mo-

tivates a generalisation to an arbitrary Riemannian manifold M , using maps x : [t0, t1]→M which are critical points of the functional∫ t1

t0

〈∇tx(1)(t),∇tx(1)(t)〉dt (2.3)

subject to boundary conditions x(t0) = x0, x(t1) = x1, x(1)(t0) = A0, x

(1)(t1) = A1, where

x0, x1 ∈ M and A0, A1 are tangent vectors at x0, x1. Solutions of the corresponding

Euler-Lagrange equation are called Riemannian cubics. We discuss the literature on Rie-

mannian cubics in Section 2.3. Higher covariant derivatives can be used in (2.3) to give a

generalisation of odd-degree polynomials.

Another generalisation is to modify the equation x(4)(t) = 0 by replacing derivatives

past the first with covariant derivatives:

∇3tx

(1)(t) = 0. (2.4)

These are Jupp and Kent’s cubics or JK-cubics, and are discussed in Section 2.5. An

example of a JK-cubic in the hyperbolic plane is shown in Figure 2.1. other powers of ∇tcan be used in (2.4) to give a generalisation of polynomials of other degrees.

There are several other generalisations which are not considered in this thesis. The de

Castljau algorithm for computing Bezier curves is generalised to splines in Riemannian

manifolds. This was done for the 3-sphere in [78] and studied in spheres, the rotation

group and other Riemannian manifolds and Lie groups in [18, 26, 61, 67, 68].

There are other alternatives which depend on more structure than an affine connection

or Riemannian metric. For a manifold embedded in Euclidean space, [33, 72] propose

interpolation using critical curves x : [t0, t1]→M of∫ t1

t0

〈x(2)(t), x(2)(t)〉dt,

21

Figure 2.1: A JK-cubic in the hyperbolic plane, drawn on the Poincare disc.

subject to endpoint constraints. Buss and Fillmore [12] propose interpolation in spheres

using weighted spherical averages of the data points with time-varying weights. The theory

of rational splines in the 6-dimensional group SE(3) of orientation-preserving isometries

of Euclidean 3-space is reviewed in [40, Sections 1–2] and [74].

We mention also the possibility of interpolation using elastic curves. The geodesic

curvature of a curve x in a Riemannian manifold is the function k : t → ‖∇tx(1)(t)‖,the norm of the rate of change of the unit tangent vector. An elastic curve is a critical

point of the functional∫ t1t0k(t)2dt, subject to given endpoints and end unit tangent vectors.

Solutions are known in certain manifolds of constant curvature [39] and in SO(3) [69] where

they are proposed for trajectory planning for rigid body motion. Closed elastic curves are

studied in [10, 45] and other variational problems based on curvature are studied in, for

example, [3–5, 8]. Elastic curves have very different properties from Riemannian cubics.

2.3 Riemannian cubics

Interpolation using curves which are extremals of (2.3) was first proposed by Gabriel

and Kajiya in [24]. They were particularly interested in the 2-sphere and the rotation

group SO(3), for applications in computer graphics and robotic control. They found the

Euler-Lagrange Equation, and gave a numerical method for its solution in the sphere.

Theorem 2.1. Let M be a Riemannian manifold with Levi-Civita connection ∇. Smooth

critical points of (2.3) with boundary conditions

x(t0) = x0, x(t1) = x1, x(1)(t0) = A0, x

(1)(t1) = A1

satisfy the Euler-Lagrange equation

∇3tx

(1)(t) +R(∇tx(1)(t), x(1)(t))x(1)(t) = 0, (2.5)

Proof. Let x : [t0, t1] × (−ε, ε) → M denote a variation of curves such that x(t0, u) =

x0, x(t1, u) = x1, xt(t0, u) = A0, xt(t1, u) = A1. Let J(u) =∫ t1t0〈∇txt,∇txt〉dt. The curve

x(t, 0) is a critical point of J if, for every variation, dJ/du = 0. Using the fact that ∇ is

metric compatible and torsion free, and that [xt, xu] ≡ 0,

dJ

du= 2

∫ t1

t0

〈∇u∇txt,∇txt〉dt

= 2

∫ t1

t0

〈∇t∇txu +R(xu, xt)xt,∇txt〉dt. (2.6)

22

Using (1.11) the second term of (2.6) becomes

〈R(xu, xt)xt,∇txt〉 = 〈R(∇txt, xt)xt, xu〉. (2.7)

Integrating the first term of (2.6) by parts,∫ t1

t0

〈∇t∇txu,∇txt〉dt = [〈∇txu,∇txt〉]t1t0 −∫ t1

t0

〈∇txu,∇2txt〉dt.

But, since ∇ is torsion-free we have [〈∇txu,∇txt〉]t1t0 = [〈∇uxt,∇txt〉]t1t0 which is zero

because xt|t=t0 = A0 and xt|t=t1 = A1 do not depend on u. Integrating by parts again,

−∫ t1

t0

〈∇txu,∇2txt〉dt = −[〈xu,∇2

txt〉]t1t0

+

∫ t1

t0

〈xu,∇3txt〉dt. (2.8)

The term −[〈xu,∇2txt〉]

t1t0

is zero because x|t=t0 = x0 and x|t=t1 = x1 do not depend on u.

Therefore combining (2.6), (2.7), (2.8) we have∫ t1

t0

〈∇3txt +R(∇txt, xt)xt, xu〉dt = 0

for any variation. By the fundamental lemma of the calculus of variations, (2.5) follows.

As pointed out in [65, Section 1.3], critical points of (2.3) are solutions of (2.5) even in

manifolds with a semi-Riemannian metric. In the case of a complete Riemannian manifold,

there always exists a minimising Riemannian cubic for given boundary conditions [27],

in the following sense: Given points p1, p2 and tangent vectors A1 ∈ Tp1M,A2 ∈ Tp2M ,

there exists a curve which is a global minimum of (2.3) subject to the constraints x(ti) =

pi, x(1)(ti) = Ai. [27] also prove Palais-Smale conditions and results on the multiplicity of

solutions. A theory of Jacobi fields for Riemannian cubics is studied in [14].

Riemannian cubics were proposed independently by [38], who also proposed the JK-

cubics piecewise satisfying (2.4). [57] also independently proposed Riemannian cubics and

studied them in SO(3) with a bi-invariant Riemannian metric (as described in Section

1.9). They considered the left Lie reduction V : R→ so(3) (see Section 1.8),

V = (dLx(t)−1)x(t)x(1)(t) (2.9)

and found that V satisfies the equation

V (3)(t) = [V (2)(t), V (t)] (2.10)

which can be integrated once to give

V (2)(t) = [V (1)(t), V (t)] + C (2.11)

where C ∈ so(3) is an arbitrary constant. As explained in Section 1.9, the Lie algebra

so(3) can be treated as R3 with the cross product for its bracket operation.

Equations (2.10) and (2.11) were generalised in [19] to other compact Lie groups. In

fact let G be a Lie group with Lie algebra g and let ∇ be the (0)-connection on G. From

Equations (1.20), (1.23) we can compute that a curve x : R→ G satisfies (2.5) if and only

23

if its Lie reduction V : R → g is a solution of (2.11). Solutions V of (2.11) are called

Lie quadratics. A related differential equation holds for Riemannian cubics in arbitrary

Riemannian symmetric spaces [19].

A null Lie quadratic in a Lie algebra g is a solution V : R → g of (2.11) with C = 0.

Null Lie quadratics in so(3) were studied in [51]. For a null Lie quadratic, curvature is

constant and torsion is a linear function of time. After a time-shift and scaling, the norm

‖V ‖ is of the form√t2 + d0 for some constant d0. The curve V has axes

limt→±∞

V

‖V ‖

and V converges to its axes as t→ ±∞. A Riemannian cubic x associated to V via (2.9)

is called a null Riemannian cubic. It has asymptotic geodesics, i.e. there are geodesics

γ+, γ− in SO(3) such that

limt→∞

d(x(t), γ+) = limt→−∞

d(x(t), γ−) = 0.

More precise asymptotics for null cubics and Lie quadratics are given in [56].

Non-null Lie quadratics and Riemannian cubics have more complex asymptotics. They

are studied in SO(3) in [52]. In SO(3), a Riemannian cubic x and a Lie quadratic V can

always be extended indefinitely to R [52, Theorem 2 and Corollary 5]. There exist Lie

quadratics which are periodic but non-constant [52, Example 6]. For a large class of Lie

quadratics, ‖V ‖ grows asymptotically as t2 (as for the majority of cubics in Euclidean

space), and V has asymptotic axes.

The dual of a Lie quadratic was introduced in [53]: if x is a Riemannian cubic in G,

with Lie quadratic V in g, then the pointwise inverse t 7→ x(t)−1 is also a Riemannian

cubic. Writing V ∗ for the Lie quadratic corresponding to t 7→ x(t)−1, we find

V ∗(t) = −Ad(x(t))V (t).

Since left and right multiplication by any element of g are affine maps, t 7→ gx(t) and

t 7→ x(t)g are Riemannian cubics whenever x is. Furthermore the time-shift x(t− c) of a

cubic x(t) is also a cubic. Therefore, given any cubic x1 we can replace it with a cubic x

such that the domain contains 0 and x(0) = 1. In this case, [53, Theorem 4], the dual V ∗(t)

satisfies V ∗(2)(t) = [V ∗(1)(t), V (t)] +C∗ where C∗ = −V (2)(0). The dual of a constant Lie

quadratic t 7→ V0 is a constant. The dual of a null Lie quadratic is an affine Lie quadratic,

V ∗ : t 7→ V0 + tV1.

The cubic corresponding to an affine Lie quadratic is called affine. Noakes [56] has related

affine Riemannian cubics to the motion of a spherically symmetric object under constant

torque; and the motion of a spherically symmetric ball rolling on a tilted plane.

With various choices of left- and bi-invariant semi-Riemannian metrics, Riemannian

cubics in SE(3) are studied in for example [77, 85] for applications in robotics.

Recently, a theory of higher order Euler-Poincare reduction has been developed [25],

and the relation between the variational problem (2.3) and Equation (2.10) is a special

case. A large class of variational problems on Lie groups can be treated with this theory.

In [25], Riemannian cubics were suggested for use in the differentiable interpolation of

image data.

24

2.3.1 Lax constraints

In this section we discuss the question of solving a Riemannian cubic x (satisfying, say,

x(0) = 1) from its Lie quadratic V . This was first done in SO(3) and SO(1, 2) in [53], and

generalised to apply to Riemannian cubics in tension and elastic curves (see Section 2.4.1)

in [59]. But [54] gave a much more general argument applying to arbitrary semisimple Lie

groups.

Let g be a Lie algebra and let V : R→ g. A Lax constraint on V is a curve Z : R→ g

such that the pair Z, V satisfy the Lax equation

Z(1)(t) = [Z(t), V (t)] (2.12)

If G is a Lie group with Lie algebra g and x : R→ G is a solution of dLx(t)−1x(1)(t) = V (t)

with x(0) = 1, then

Z(t) = Ad(x(t))Z(0),

so Z(t) is the image of Z(0) under an isomorphism of the Lie algebra. If g is a matrix

Lie algebra, it follows that Z(t) is isospectral, i.e the characteristic polynomial of Z(t) is

constant.

Noakes found in [54] that if g is a semisimple Lie algebra, there is an open dense subset

g(1) of g with the following property: if Z(t) is a Lax constraint with Z(0) ∈ g(1), then x(t)

can be solved in terms of V,Z using algebraic operations and quadrature (integration).

Since g(1) is an open dense set, the condition that Z(0) ∈ g(1) is generic. (In Chapter

3 we will give more detail and generalise the above result to Lie algebras that are not

semisimple.)

The usefulness of this for Riemannian cubics is that when V satisfies (2.11), there is a

Lax constraint

Z(t) = V (2)(t).

Thus, under generic initial conditions on V , the Riemannian cubic x can be solved from

V, V (2) in quadrature. Noakes also found an open dense subset g(2) of g × g such that if

Z1, Z2 are Lax constraints with (Z1(0), Z2(0)) ∈ g(2) then x can be solved algebraically

from V,Z1, Z2. If V is a null Lie quadratic then

Z1 = V (1)(t) Z2 = V (t)− tV (1)(t)

are Lax constraints. Therefore under generic conditions on V , the Riemannian cubic x

can be solved algebraically in terms of V, V (1).

2.4 Generalisations of Riemannian cubics

2.4.1 Riemannian cubics in tension

Critical points of the functional∫ t1

t0

(〈∇tx(1)(t),∇tx(1)(t)〉+ τ〈x(1)(t), x(1)(t)〉

)dt, τ > 0 (2.13)

25

satisfy the Euler-Lagrange equation

∇3tx

(1)(t) +R(∇tx(1)(t), x(1)(t))x(1)(t)− τ∇tx(1)(t) = 0. (2.14)

Solutions of (2.14) are called Riemannian cubics in tension (RCTs). The constant τ is

called the tension parameter. RCTs are a compromise between Riemannian cubics and

geodesics. They are appropriate for interpolation problems where short paths are needed.

RCTS were proposed in [79] and [80] where the Lie reduction V of an RCT x is shown to

satisfy

V (2)(t) = [V (1)(t), V (t)] + τV (t) + C (2.15)

for a constant C lying in the Lie algebra. Solutions of (2.15) are studied in [58] where they

are called Lie quadratics in tension (LQTs). An LQT is called null if C = 0. [58] gives

asymptotics for null LQTs in so(3). Corresponding to null LQTs are null Riemannian

cubics in tension and like null Lie quadratics (without tension), null Lie quadratics in

tension in SO(3) have asymptotic geodesics [58, Theorem 8]. Riemannian cubics in tension

are also studied in [6, 36]

Critical points of (2.13) are called elastic curves in [79], but elsewhere (e.g. [65]) they

are called Riemannian cubics in tension to avoid confusion with the generalization of Euler

elastic curves to Riemannian manifolds (on page 22).

2.4.2 Higher order geodesics

Critical points of the functional∫ t1

t0

〈∇m−1t x(1)(t),∇m−1t x(1)(t)〉dt (2.16)

were first proposed in [13] to produce splines of class C2m−2. The curves are variously

called Riemannian polynomials [28], curves of class Dm [42] or m-geodesics [66].

[13] gave the Euler-Lagrange equation for (2.16), and [28] investigate analytic aspects

including the Palais-Smale conditions. In the case m = 2, Equation (2.16) becomes (2.3).

Krakowski [42, Section 5.3] considered the case m = 3 in the Lie group SO(3), found

a fifth order ordinary differential equation for the Lie reduction, and integrated it to a

fourth order equation [42, Corollary 5.3.8] (c.f. the integration of (2.10) to give (2.11)).

Popiel [65, Section 1.3] points out that Krakowski’s proof applies in any Lie group with a

bi-invariant semi-Riemannian metric. In [66], Popiel treats (2.16) for arbitrary m, giving

a 2m − 1 order ordinary differential equation for the Lie reduction V of a solution x of

(2.16), and integrates it to a 2m − 2 order ODE. Popiel also finds a Lax constraint in

terms of V and its derivatives. Thus by the theory discussed above in Section 2.3.1, under

generic initial conditions, the curve x can be solved by quadrature in terms of V and its

derivatives.

Riemannian cubics and 3-geodesics are studied in SE(3) in [86].

2.5 Jupp and Kent’s cubics

Jupp and Kent proposed in [38] to produce smoothing splines in the sphere using piece-

wise solutions of (2.4). A geometric interpretation is as follows: Given a piecewise smooth

26

curve in the 2-sphere, one can roll the sphere on a plane, along the curve, without slipping

or twisting. The point of contact traces out a curve in the plane, sometimes called the de-

velopment of the original curve. A curve in the sphere satisfies (2.4) when its development

in the plane is a cubic (from [38, Equation 2.2]. In fact [38, Section 1] pointed out that

their definition would apply to other Riemannian manifolds such as higher dimensional

spheres, but their exposition stuck to the unit sphere. The applications of interest were

vectorcardiograms, and apparent polar wander paths.

Hanna and Chang in [30] propose a modification of Jupp and Kent’s method for in-

terpolation in the 3-sphere (treated as the group of unit quaternions) and SO(3). These

Lie groups have a bi-invariant Riemannian metric. The Levi-Civita connection is the (0)-

connection but [30] suggests that for the interpolation of tectonic plate motion, it may be

better to use curves x for which x−1x(1) is piecewise cubic. In other words, [30] suggests

interpolation using JK-cubics with the (-)-connection.

Kume, Dryden and Le [43] used Jupp and Kent’s cubics for smoothing splines in the

shape space of k points in 2 dimensions. This manifold is constructed as follows: start

with the set of all k-tuples (p1, · · · , pk) of points in R2 such that not all pi are equal. Let

two elements of this set be equivalent if they differ by an orientation-preserving similarity.

Shape space is the set of all equivalence classes. Kume, Dryden and Le give a method

for numerically approximating Jupp and Kent’s cubics and apply it to human movement

data.

The technique of rolling manifolds on Euclidean space is used in other approaches to

interpolation in curved space [35].

For a generalisation of higher order polynomials one might consider solutions of

∇m−1t x(1)(t) = 0.

Larsen [46] studies these curves and calls them m geodesics (although in the present

thesis the term is used for the higher order generalisations in Section 2.3). Larsen studies

variations through these curves and develops a concept of m Jacobi fields.

2.6 Overview of results in this thesis

The use of Lax constraints, discussed above in Section 2.3.1, to solve a curve by quadrature

from its Lie reduction, was developed for particular Lie groups in [53, 59] and generalised

to semisimple Lie groups in [54]. In Chapter 3 we generalise this result to all Lie groups.

In Chapter 4 we study the equation

∇2tx

(1)(t) = 0 (2.17)

in Lie groups. Equation (2.17) is a special case of Equation (2.4). We call the solutions

JK-quadratics. The JK-quadratics in a Lie group can be solved in terms of the null Lie

quadratics in the corresponding Lie algebra. In SO(3) and SO(1, 2) and their corresponding

Lie algebras, we solve for null Lie quadratics and JK-quadratics in terms of solutions of

the differential equation for a quantum harmonic oscillator.

27

In Chapter 5 we study JK-cubics in matrix groups, and in particular the general linear

group GL(n). The ordinary differential equation (2.4) for a JK-cubic x in a matrix group

with the (0)-connection can be reduced to a pair of linear systems

σ(1)0 (t) = σ0(t)(F0 + F1t+ F2t

2) ρ(1)0 (t) = (F0 + F1t+ F2t

2)ρ0(t)

where F0, F1, F2 are elements of the corresponding matrix Lie algebra, depending on the

initial conditions of x. When the leading coefficient F2 has distinct eigenvalues, the theory

of asymptotic expansions of solutions of linear systems [84] can then be used to easily get

an asymptotic description of the JK-cubic. The n × n matrices with distinct eigenvalues

form an open dense subset of the vector space of all n × n matrices. Thus, we give

asymptotic expansions for almost all JK-cubics in GL(n).

In Chapter 6 we consider the possibility that the Lie algebra g of a Lie group G may

not have elements with distinct eigenvalues. The easy case of [84, Chapter IV] does not

apply – asymptotic expansions are possible, but more difficult to find [84, Chapter V]. We

find that there is an open dense subset of g such that, when the leading coefficient of a

linear system lies in this open dense subset, the asymptotic solutions can be found easily

with a similar approach to [84, Chapter IV].

In Chapter 7 we study the question of extendibility of Riemannian cubics. In a com-

plete Riemannian manifold M , part of the Hopf-Rinow theorem tells us that geodesics

are extendible indefinitely, namely, any geodesic is the restriction to its domain of some

geodesic R → M . We find that this is not true for Riemannian cubics. In particular,

say a manifold has strictly negative sectional curvature if there is some λ > 0 such that

K(σ) < −λ for every plane σ at every point. In any manifold of strictly negative sectional

curvature, we find an open family of Riemannian cubics all of whose velocities diverge in

finite time. Thus they cannot be extended.

In Chapter 8, with the goal of giving an asymptotic description of JK-cubics in a

manifold M of strictly negative curvature, we generalise a result of [23]. For a curve

y : R → M parametrised by arc-length, the geodesic curvature of x is defined as k(t) =

‖∇ty(1)(t)‖. In [23] it is proved that for a curve y in the hyperbolic plane, if, for some

t0 ∈ R,∫∞t0k(t)dt is finite, then there is a geodesic γ such that d(y(t), γ) converges to zero

as t→∞. We generalise this result to arbitrary manifolds of strictly negative curvature.

JK-cubics satisfy this condition, which means that any JK-cubic in a manifold of strictly

negative curvature can be asymptotically approximated by a (reparametrised) geodesic.

28

Chapter 3

Lax constraints in arbitrary Lie groups

Let G be a connected, nontrivial, real or complex Lie group with Lie algebra g. For a

curve x : R→ G, there is a corresponding curve V : R→ g, the Lie reduction of x, given

by

V (t) = dLx(t)−1x(1)(t).

A Lax constraint on V is a curve Z : R→ g such that

Z(1)(t) = [Z(t), V (t)].

A prototype for x is any curve y : R→ G such that Ad(y(t))Z(t) is a constant D ∈ g, and

y(t0) = x(t0) for some t0. The curve x is a prototype for itself but other prototypes can

be found algebraically from Z, as explained in [54].

Recall that the rank of g is the smallest integer r such that the map

X 7→ coefficient of λr in the characteristic polynomial of X

is not identically zero. Write η(X) for the coefficient of λr. An element X of g is called

regular if η(X) is nonzero. An argument using Zariski-open sets shows that almost all

elements of g are regular.

If D is regular, then x(t) can be solved by quadrature in terms of V (t), Z(t), y(t) and

the exponential map exp : g → G. Noakes proved this for semisimple G in [54] and the

purpose of this chapter is to modify the proof to apply to arbitrary G.

In [53, 54] this is shown to be useful for the study of Riemannian cubics in a Lie group

with a bi-invariant semi-Riemannian metric. Let V be the Lie reduction of a Riemannian

cubic x. From Equation (2.10), we see that V (2) is a Lax constraint for V . Thus, for

generic initial conditions on V , the curve x can be found by quadrature from V, V (2) and

a prototype. The integral (2.3) only makes sense when G has a semi-Riemannian metric,

and (2.10) only follows when the semi-Riemannian metric is bi-invariant. But even in this

case, initial conditions of (2.10) may lie in some Lie subalgebra m of g which does not have

a bi-invariant semi-Riemannian metric. Then D might be a regular element of m without

being a regular element of g, so that the generalisation to arbitrary Lie algebras (rather

than just to those with bi-invariant semi-Riemannian metrics) is of assistance.

For a Riemannian cubic in tension (critical points of the functional given in (2.13)), the

Lie reduction V satisfies Equation (2.15) and [58] there is a Lax constraint Z(t) = V (2)(t)−

29

τV (t). The geodesics of order k defined variationally by (2.16) have Lax constraints which

were constructed for every k by [66].

From Section 1.7, if X ∈ g is a regular element then the set

h = Y ∈ g : (adX)sY = 0 for some integer s ≥ 1 (3.1)

is a Cartan subalgebra of g and is therefore nilpotent. The dimension of h is equal to the

rank of g. There are ideals h0 = h ⊇ h1 ⊇ · · · ⊇ hr = 0 such that dim hi = r − i, and

[h, hi] ⊆ hi+1. Choose a constant basis Uir−1i=0 of h such that Ui ∈ hi\hi+1. Then for any

Y ∈ h and j ≥ 1 we have ad(Y )jUi ∈ hi+j+1 ⊆ hi+1 and ad(Y )r−iUi = 0 so that

Ad(exp(Y ))Ui = exp(ad(Y ))Ui = (1 + ad(Y ) + · · ·+ 1

(r − i− 1)!ad(Y )r−i−1)Ui)

∈ Ui + hi+1. (3.2)

For an element Y ∈ h, write πi(Y ) for the coefficient of Ui when Y is written in the basis

U0, · · · , Ur−1.

Lemma 3.1. Let v be a curve in a Cartan subalgebra h of g. Then we can solve the

differential equation

dLh(t)−1h(1)(t) = v(t)

h(t0) = 1(3.3)

by quadrature, as follows:

h(t) = exp(

∫ t

t0

fr−1(τ)dτ Ur−1) · · · exp(

∫ t

t0

f0(τ)dτ U0) (3.4)

where we solve fi recursively by letting v0(t) := v(t) and

fi(t) = πi (vi(t))

vi+1(t) = Ad(

exp(∫ tt0fi(τ)dτ Ui)

)(vi(t)− fi(t)Ui(t)) .

(3.5)

Proof. Write (3.4) for unknown functions f0, · · · , fr−1. Then, differentiating (3.4) and

using (3.3), we have v(t) = dLh(t)−1h(1)(t) =

(f0(t)U0)

+ Ad

(exp(−

∫ t

t0

f0(τ)dτ U0)

)(f1(t)U1)

+ · · ·

+ Ad

(exp(−

∫ t

t0

f0(τ)dτ U0) · · · exp(−∫ t

t0

fr−2(τ)dτ Ur−2)

)(fr−1(t)Ur−1). (3.6)

By Equation (3.2), at any given time t, term i in this expression lies in the coset fi(t)Ui +

hi+1. Of course, term i + 1 is contained in fi+1(t)Ui+1 + hi+2 ⊆ hi+1. So we can solve

Equation (3.3) step by step for the unknown functions fi, using (3.5).

Assuming we have a Lax constraint Z, we can now solve x from V,Z, y. The proof

is as in [54, Section 4] except that we use the method above in the case that the Cartan

subalgebra is not abelian.

30

Theorem 3.2. Given V (t), Z(t), y(t), suppose D is regular. Let

v(t) = −Ad(y(t))(dLy(t)−1y(1)(t)− V (t)). (3.7)

Then the solution x of

dLx(t)−1x(1)(t) = V (t)

x(t0) = y(t0)(3.8)

is

x(t) = exp(

∫ t

t0

fr−1(τ)dτUr−1) · · · exp(

∫ t

t0

f0(τ)dτU0)y(t),

where U0, · · · , Ur−1 is a basis for the Cartan subalgebra containing D, and f0, · · · , fr−1are computed from (3.5) with v0 = v.

Proof. Let h(t) = x(t)y(t)−1. Then since Ad(x(t))Z(t) = D = Ad(y(t))Z(t) we have

Ad(h(t))D = D for all t. Therefore

Ad(h(t))Ad(h(t)−1h(s))D = D for all s, t. (3.9)

Let v(t) be the left Lie reduction of h(t). Then, differentiating (3.9) in s and evaluating

at s = t, ((d(Ad(h(t)))1 (dAd)1

(dLh(t)−1

)h(t)

d

ds|s=t(h(s))

)D = 0,

so that ad(v(t))D(= [v(t), D]) = 0. Since D is regular, v(t) lies in the Cartan subalgebra

given by Equation (3.1). Applying Lemma 3.1, we can solve h(t) from v(t). To see that

v(t) is given by (3.7), use v(t) = dLh(t)−1h(1)(t) and h(t) = x(t)y(t)−1:

v(t) = dL(x(t)y(t)−1)−1

d

dt

(x(t)y(t)−1

)= dLy(t)dLx(t)−1

(dLx(t)

d

dt

(y(t)−1

)+ dRy(t)−1

d

dt(y(t))

)= dLy(t)dLx(t)−1

(dLx(t)

(−dLy(t)−1dRy(t)−1y(1)(t)

)+ dRy(t)−1x(1)(t)

)= −dRy(t)−1y(1)(t) + Ad(y(t))dLx(t)−1x(1)(t)

= −Ad(y(t))(

dLy(t)−1y(1)(t) + V (t)).

31

32

Chapter 4

Jupp and Kent’s quadratics in Lie groups1

Abstract

To interpolate a sequence of points in Euclidean space, parabolic splines

can be used. These are curves which are piecewise quadratic. To interpolate

between points in a (semi-)Riemannian manifold, we could look for curves such

that the second covariant derivative of the velocity is zero. We call such curves

Jupp and Kent quadratics or JK-quadratics because they are a special case of

the cubic curves advocated by Jupp and Kent. When the manifold is a Lie

group with bi-invariant metric, we can relate JK-quadratics to null Lie quadrat-

ics which arise from another interpolation problem. We solve JK-quadratics in

the Lie groups SO(3) and SO(1, 2) and in the sphere and hyperbolic plane, by

relating them to the differential equation for a quantum harmonic oscillator.

4.1 Introduction

Suppose a sequence of times t1, · · · , tn and a sequence of points x1, · · · , xn are given, and

a curve x is desired which is sufficiently smooth and satisfies x(ti) = xi. When the xi are

in Euclidean space, one method of producing such curves is the natural cubic spline which

is a solution of

x(4) = 0 (4.1)

whenever ti < t < ti+1, chosen so that the curve is C2 at each of t2, · · · , tn−1, and

x(2)(t1), x(2)(tn) are both 0. Natural cubic splines are extrema of the functional∫ tn

t1

〈x(2), x(2)〉dt.

They have motivated several generalisations to arbitrary Riemannian manifolds. Two of

them are produced by replacing ddt by the Levi-Civita covariant derivative ∇t.

1Published as [64]. To have notation consistent with the rest of the thesis, Dots for derivatives have

been replaced with (1), (2) etc., and ∇d/dt has been replaced with ∇t. Some of the figures have been

improved and a couple of minor changes have been made to the exposition.

33

• Riemannian cubics: Equation (4.1) is the Euler-Lagrange equation for the La-

grangian 〈x(2), x(2)〉. Replacing this by 〈∇tx(1),∇tx(1)〉 gives the Euler-Lagrange

equation [57]

∇3tx

(1) +R(∇tx(1), x(1))x(1) = 0

(where R is the Riemannian curvature tensor);

• JK-cubics A simpler ODE can be formed by replacing d/dt by ∇t directly in (4.1):

∇3tx

(1) = 0. (4.2)

We are calling such curves Jupp and Kent cubics or JK-cubics after the authors of

[38] who first suggested these curves for approximation in the 2-sphere.

If there is no reason to believe that x should be C2, we might have (in the Euclidean

case) used parabolic splines [20, Chapter VI], which have less degrees of freedom, and can

only be chosen to be C1 at the times ti. They satisfy

x(3) = 0 (4.3)

whenever t is not one of the ti, but they do not appear to solve any natural variational

problem. Therefore we do not know of a variational way of generalising (4.3), but we

can directly replace ddt by ∇t in the equation (4.3), giving what we call JK-quadratics

satisfying

∇2tx

(1) = 0. (4.4)

In Section 4.2, we consider (4.4) in Lie groups with a bi-invariant (semi-)Riemannian

metric (and corresponding Levi-Civita connection). In this case we can reduce the equation

to a differential equation in the Lie algebra:

V (2) = [V (1), V ]. (4.5)

This is the equation for a null Lie quadratic, which actually first arose in a special case

of Riemannian cubics in a Lie group [51]. In general we solve (4.4) in terms of a pair of

solutions to (4.5). (However it turns out in Section 4.3 that in the particular Lie groups

SO(3) and SO(1, 2) we can solve (4.4) in terms of a single solution of (4.5).) In Section

4.3 we review some properties we need of the Lie groups SO(3) and SO(1, 2) and their Lie

algebras.

In Section 4.4 we give a solution for (4.5) in the Lie algebra sl(2,C). Using the duality

of Lie quadratics [53], the equation (4.5) in sl(2,C) can be solved in terms of the solutions

y : C→ SL(2,C)

of

y(1) = (A0 +A1t)y (4.6)

where A0, A1 are constant elements of sl(2,C). We solve this equation in sl(2,C) in terms

of solutions to the time-independent Schrodinger equation with quadratic potential. Since

34

sl(2,C), so(1, 2,C) and so(3,C) are isomorphic, this gives the solution of null Lie quadratics

in so(1, 2,C) and so(3,C).

For the Lie algebra so(3,R), the equation (4.6) has already been solved by Romano[73],

using a stereographic projection to transform it into a Riccati equation, and then trans-

forming the Riccati equation into a second order linear ODE. A similar approach is taken

in [29, Section 3] to solve Euler spirals on the sphere which relate to the self-similar motion

of vortex filaments, and which turn out to be the same thing as V (1) in (4.5). [21] studies

an analogous problem in Minkowski 3-space, and in Section 3 uses a similar method to

solve self-similar motion. Such solutions could be thought of as Euler spirals on the hyper-

bolic plane. By working in sl(2,C) we can be slightly more direct than these approaches.

However, in sl(2,C) there is a degenerate case which we must take into account (which

does not show up in the above situations).

In Section 4.8 we consider (4.4) in the sphere and the hyperbolic plane. It is shown

that the solutions of (4.4) can be computed from null Lie quadratics in sl(2,C).

4.2 JK-quadratics in Lie groups

Suppose that G is a finite dimensional Lie group with a bi-invariant semi-Riemannian

metric. Let ∇ be the Levi-Civita connection corresponding to this metric. In studying

(4.4) we may assume that x(0) = 1. For any g ∈ G define

Lg : G→ G Rg : G→ G

x→ gx x→ xg.

Then, from a curve x : R → G, we can form the curves W and W ∗ in the corresponding

Lie algebra g,

W (t) = dLx(t)−1

(x(1)(t)

)W ∗(t) = −dRx(t)−1

(x(1)(t)

).

(Then W ∗(t) = −Ad(x(t))W (t).)

Suppose that x satisfies (4.4). Now, using methods from [51], we have

Proposition 4.1. W satisfies the equation

W (2) +1

2[W,W (1)] = 0. (4.7)

Proof. For a vector field X(t) defined along x(t), let X(t) = dLx(t)−1X(t). For each

t we can consider X(t) as a left invariant vector field on G. Since ∇ is bi-invariant,

for any two left invariant vector fields Y,Z we have ∇Y Z = 12 [Y,Z]. It follows that

dLx(t)−1∇tX = X(1) + 12 [W,X]. Applying this to (4.4) gives (4.7).

Thus 12W is a null Lie quadratic (see Equation (4.5)). Similarly W ∗ satisfies the same

equation and 12W

∗ is a null Lie quadratic. Our present goal is to solve x from W and W ∗,

that is, to solve the initial value problem

x−1x(1) = W (4.8)

x(0) = 1. (4.9)

35

Note that for arbitrary Y : R→ g, we have ddt(Ad(x)Y ) = Ad(x)(Y (1) + [W,Y ]), thus

−Ad(x)W = W ∗ (4.10)

−Ad(x)W (1) = W ∗(1). (4.11)

We have

Proposition 4.2. If G is semisimple then in a generic case we can solve Ad(x) alge-

braically from W and W ∗.

Proof. From (4.10),(4.11) and since Ad(x) is an automorphism of g, it maps sums of

brackets of W and W (1) into corresponding sums of brackets of −W ∗ and −W ∗(1). It has

been shown ([54, Section 5]) that the pairs of elements of g which generate g form an open

dense subset of g× g. (An open-closed argument shows that the Lie algebra generated by

W (t),W (1)(t) does not depend upon t.) Thus, generically, W,W (1) generate g. We can

now produce enough linear equations to solve for Ad(x).

4.3 SO(3) and SO(1, 2)

Let us consider two particular Lie groups. (See [34, 83], for example, for details on the

structure of these groups and their corresponding Lie algebras.)

• SO(3) is the group of 3× 3 real matrices x satisfying xTx = 1. Its Lie algebra so(3)

consists of all Ω satisfying Ω + ΩT = 0. The Lie bracket [Ω1,Ω2] is the commutator

Ω1Ω2 − Ω2Ω1. There is an inner product 〈·, ·〉 which is positive definite and ad-

invariant : 〈Ω1, [Ω1,Ω2]〉 = 0. (Choose the negative of the Killing form.)

The Lie algebra so(3) is isomorphic to E3, which is R3 with the cross product as its

Lie bracket, as follows: pqr

↔ 0 q −r−q 0 p

r −p 0

.

• Let J be the 3×3 diagonal matrix with entries −1, 1, 1. The group SO(1, 2) consists

of all 3× 3 matrices x satisfying xTJx = J . The Lie algebra so(1, 2) consists of all

Ω such that ΩTJ = −JΩ. Again the Lie bracket is the commutator. Here there also

exists an ad-invariant symmetric bilinear form 〈·, ·〉, which is nondegenerate but not

positive definite.

The Lie algebra so(1, 2) is isomorphic to L3, which is R3 with Lie bracket given by

[u, v] = J(u× v), via the correspondencepqr

↔0 q r

q 0 −pr p 0

.

36

We also consider the Lie group SL(2,C) of all 2× 2 complex matrices of determinant

1. Its Lie algebra sl(2,C) consists of the 2 × 2 complex matrices of trace 0, with bracket

operator the commutator. Letting 〈X,Y 〉 = 2tr(XY ) we have

[[X,Y ], Z] = − (〈X,Z〉Y − 〈Y, Z〉X) . (4.12)

If g is one of so(3) or so(1, 2), then any X,Y, Z ∈ g satisfy the identity

[[X,Y ], Z] = ε (〈X,Z〉Y − 〈Y, Z〉X) (4.13)

where ε = 1 for so(3) and ε = −1 for so(1, 2).

In SO(3) and SO(1, 2) we can do better than Proposition 4.2:

Proposition 4.3. Let G be either of SO(3) or SO(1, 2). Then, in a generic case, W ∗ can

be found algebraically from W and thus x can be found algebraically from W alone.

Proof. As in section 4.2 we assume that 0 is in the domain of x and that x(0) = 1. For

an open dense set of initial conditions W (0),W (1)(0), the plane spanned by W (0),W (1)(0)

is orthogonal to some U ∈ g with 〈U,U〉 6= 0. Define the map

φ : g→ g

Z 7→ 2〈U,Z〉〈U,U〉

U − Z.

Now φ is linear and a calculation (using (4.13) twice) shows that [φZ1, φZ2] = φ[Z1, Z2].

Therefore φ is an automorphism, and since W (0),W (1)(0) are orthogonal to U we have

φ(W (0)) = −W (0) and φ(W (1)(0)) = −W (1)(0).

From (4.10) and (4.11) we have W ∗(0) = −W (0) = φ(W (0)) and

W ∗(1)(0) = −W (1)(0) = φ(W (1)(0)).

Since W ∗(t) and φW (t) are solutions of (4.7) with the same initial conditions, we have

W ∗(t) = φW (t). We have solved W ∗ from W , and by Proposition 4.2 we can find x in

terms of W .

4.4 Null Lie quadratics and the Schrodinger equation

Let U ⊆ C be connected and let V : C→ sl(2,C) be a solution of

V (2) + [V, V (1)] = 0,

that is, V is a null Lie quadratic in sl(2,C) (parametrised by C). Suppose that y : U →SL(2,C) satisfies y−1y(1) = V . 2 Let A(t) = y(1)y−1 = Ad(y)V . Then as in [53, Section 4]

we have A(1) = Ad(y)(V (1)) and A(2) = Ad(y)(V (2) + [V, V (1)]) = 0. Thus A = A0 +A1t.

Our goal now is to find y from A. Note that if P is a constant element of SL(2,C), and

y = P−1yP , then y satisfies y(1)y−1 = (PA0P−1 + PA1P

−1t). Therefore we can assume

without loss of generality that A1 is in Jordan form, i.e.

A(t) =

(a(t) b(t)

c −a(t)

)(4.14)

2Note that generally y is not the same as x – the factor of 12

makes a difference.

37

where a, b are affine functions of t, and c is a constant. We will call y and V degenerate

when c = 0.

Proposition 4.4. The solution for degenerate y is

g

(e∫ t0 a(τ)dτ e

∫ t0 a(τ)dτ

∫ t0 b(σ)e−2

∫ σ0 a(τ)dτdσ

0 e−∫ t0 a(τ)dτ

)h

and the solution for degenerate V is

h−1

(a(t) 2a(t)

∫ t0 b(σ)e−2

∫ σ0 a(τ)dτdσ + be−2

∫ t0 a(τ)dτ

0 −a(t)

)h

where g, h are constant elements of SL(2,C).

Proof. Write yi, (i = 1, 2) for the columns of y. Then (4.14) (with c = 0) gives y(1)i2 =

−ayi2 (which can be solved by an integrating factor) and y(1)i1 = ayi1 + byi2 (which can be

solved in terms of yi2 by an integrating factor). This gives the solution for y. Now compute

V = y−1y(1) = y−1Ay. (The constants g and h take care of the conjugation required to

put A in the form (4.14), as well as translating y to an arbitrary initial position.)

We can now concern ourselves with the case when c 6= 0. The one-dimensional time-

independent Schrodinger equation for a function ψ : R→ C is

d2ψ

dq2= (v(q)− E)ψ (4.15)

where v : R→ R is the potential and E is a constant. It arises when looking for standing

waves of the time-dependent Schrodinger equation. When v(q) is a quadratic function

of q, Equation (4.15) is known as the equation of a quantum harmonic oscillator , and

the solutions are special functions that are well known (See for example [44, Section 23,

and Appendix a], in which the ODE is solved by a contour integral, and the many exact

solutions involving Hermite polynomials are found.) In our application we will be using

solutions in which v can be complex. This does not stop the equation from being solved

in terms of special functions.

Theorem 4.5. The solution for nondegenerate y is

g

1c

(ψ(1)1 + a(t)ψ1

)1c

(ψ(1)2 + a(t)ψ2

)ψ1 ψ2

h

and the solution for nondegenerate V is

1

ch−1

((−ψ(1)

1 ψ(1)2 −(ψ

(1)2 )2

(ψ(1)1 )2 ψ

(1)1 ψ

(1)2

)− (a2 + bc)

(−ψ1ψ2 −ψ2

2

ψ21 ψ1ψ2

))h (4.16)

where

• g, h are constants in SL(2,C);

38

• ψ1, ψ2 are solutions of the initial value problem

ψ(2)(t) + v(t)ψ(t) = 0 ψ1(0) = 0 ψ(1)1 (0) = c

ψ2(0) = 1 ψ(1)2 (0) = −a(0)

where v(t) is the quadratic function a(1)(t)− a(t)2 − b(t)c.

Proof. As with the case c = 0, let yi, (i = 1, 2) be the columns of y. Then (4.14) gives,

for each i,

y(1)i1 = ayi1 + byi2 (4.17)

y(1)i2 = cyi1 − ayi2. (4.18)

Differentiating (4.18) and substituting (4.17),(4.18) gives y(2)i2 + (a(1) − a2 − bc)yi2 = 0.

Then from (4.18) we can solve yi1 in terms of yi2, y(1)i2 .

(Note that cdet y is the Wronskian ψ(1)1 ψ2 − ψ1ψ

(1)2 , and in ODEs of the form (4.15)

the Wronskian is constant. Thus by the initial condition, y ∈ SL(2,C) for all t ∈ C.)

Lastly we compute V = y−1y(1) = y−1Ay.

4.5 Higher derivatives

Let ψ1, ψ2 be as in Theorem 4.5 and let

U0 =

(−ψ1ψ2 −ψ2

2

ψ21 ψ1ψ2

), U1 =

(−ψ(1)

1 ψ2 − ψ1ψ(1)2 −2ψ

(1)2 ψ2

2ψ(1)1 ψ1 ψ

(1)1 ψ2 + ψ1ψ

(1)2

),

U2 =

(−ψ(1)

1 ψ(1)2 −(ψ

(1)2 )2

(ψ(1)1 )2 ψ

(1)1 ψ

(1)2

).

The equation ψ(2)i + vψi = 0 gives the relations

U(1)0 = U1 U

(1)1 = 2U2 − 2vU0 U

(1)2 = −vU1. (4.19)

Equation (4.16) becomes

V =1

ch−1

(U2 + (v − a(1))U0

)h.

Differentiating, and using (4.19) gives

V (1) =1

ch−1

(−a(1)U1 + v(1)U0

)h (4.20)

V (2) =1

ch−1

(−2a(1)U2 + v(1)U1 + (2a(1)v + v(2))U0

)h. (4.21)

4.6 SU(2), SO(1, 2,R), SO(3,R) and their Lie algebras

We now consider SU(2), the (real Lie) subgroup of SL(2,C) consisting of those A such

that

AAT = 1. (4.22)

39

Its Lie algebra su(2) consists of those complex 2× 2 matrices Ω of trace 0 satisfying

Ω + ΩT = 0. (4.23)

We want to pick out those solutions y which lie in SU(2). This happens when su(2) 3

y(1)y−1 =

(a(t) b(t)

c −a(t)

). Therefore:

Proposition 4.6. The solutions for V in su(2) and y in SU(2) can be found by restricting

the solutions in Proposition 4.4 and Theorem 4.5 to the following conditions:

• a maps R to imaginary numbers;

• b is the constant −c;

• g, h ∈ SU(2).

Now we can find null Lie quadratics in E3 by applying the isomorphism

ψE3 : su(2)→ E3

(p q

r −p

)7→

−2ip

i(q + r)

q − r

. (4.24)

to the formulae from Proposition 4.4 and Theorem 4.5.

We can also find null Lie quadratics in L3. The solutions for V in sl(2,R) and y in

SL(2,R) can be found by restricting the solutions in Proposition 4.4 and Theorem 4.5 to

those for which a, b, c are real, and g, h ∈ SL(2,R). Now use the isomorphism

ψL3 : sl(2,R)→ L3

(p q

r −p

)7→

q − r−2p

q + r

. (4.25)

4.7 Euler spirals

An Euler spiral is a curve whose curvature is a linearly increasing function of its arc-length

parameter. For a curve x(t) parametrised with unit speed in a Riemannian manifold, the

geodesic curvature is κ(t) =√〈∇tx(1),∇tx(1)〉. Euler spirals in a sphere are related to

self-similar motion of vortices [29], and those in the hyperbolic plane are related to the

Schrodinger map for the hyperbolic plane [21].

Suppose that V is a null Lie quadratic in E3. It is known ([51, Section 3]) that 〈V, V 〉is a quadratic function of time, and 〈V (1), V (1)〉 and 〈V (2), V (2)〉 are constant. So, as

long as V is not stationary, there is some c such that Z = 1cV

(1) is a curve in the sphere

S2 = u ∈ E3 : 〈u, u〉 = 1. Furthermore the curve Z is parametrised with constant speed.

The connection arising from the restriction of the metric 〈·, ·〉 to S2 can be computed for a

vector field defined along Z by differentiating, and projecting onto TZS2. we can compute,

using both (4.5) and (4.12) several times, that

∇tZ(1) = 〈V (1), V 〉(V − 1

c〈V (1), V 〉V (1)

).

40

Figure 4.1: An Euler spiral on a sphere, and several on a hyperbolic plane, represented

as the Poincare disc.

Thus

κ =√〈∇tZ(1),∇tZ(1)〉 = 〈V (1), V 〉

√(〈V, V 〉 − 1

c2〈V (1), V 〉2

).

A further calculation shows that 〈V, V 〉 − 1c2〈V (1), V 〉2 is constant, and so κ is linearly

increasing in time. Therefore Z = 1cV

(1) is an Euler spiral in the sphere.

The same can be applied to null Lie quadratics in L3 to produce Euler spirals on

a hyperbolic plane. Here, only when 〈V (1), V (1)〉 is negative will there be c such that

Z = 1cV

(1) lies on one of the two hyperbolic planes given by u ∈ L3 : 〈u, u〉 = −1.Euler spirals on the sphere and hyperbolic plane can be computed using Proposition

4.4, Theorem 4.5 and Equations (4.20), (4.24), (4.25). Examples are given in Figure 4.1

4.8 JK-quadratics in the sphere and hyperbolic plane

Let 〈·, ·〉 be a nondegenerate symmetric bilinear form (not necessarily positive definite) on

Rn and consider its restriction to the quadric

〈x, x〉 = d (4.26)

for a constant d 6= 0. (This is required so that the restriction of 〈·, ·〉 stays nondegenerate.)

Let ∇ be the corresponding Levi-Civita connection on the quadric. What are the solutions

of ∇2tx

(1) = 0? We are interested in particular in the case when n = 3 and the quadric is

either a sphere or a two-sheeted hyperboloid (in the latter case we get the JK-quadratics in

the hyperbolic plane). (Fig. 2.) Note that the bilinear form used to define the Riemannian

metric is the same as the one used to define the quadric itself. Thus, although the bilinear

form could be chosen so that the quadric is, say, an ellipsoid, the JK-quadratics will then

be the same as those for a sphere (after a linear transformation). Thus the present method

does not produce the JK-quadratics given by the Euclidean inner product on an ellipsoid.

Let Q = 〈x(1), x(1)〉/d. Using the property that ∇ is compatible with the metric,

repeated differentiation of Q gives Q(3) = 0. Thus Q is a quadratic function of t. Differ-

41

Figure 4.2: A JK-quadratic on a sphere, and some JK-quadratics in the hyperbolic

plane (drawn on the Poincare disc).

entiating (4.26) several times, we have

0 = 〈x, x(1)〉 = 〈x, x(2)〉+ 〈x(1), x(1)〉 = 〈x, x(3)〉+ 3〈x(1), x(2)〉.

For a vector field X defined along x, ∇tX is the projection of X(1) onto the tangent space

at x of the quadric. Thus

∇tx(1) = x(2) − 〈x, x(2)〉d

x = x(2) +Qx

0 = ∇2tx

(1) = x(3) +Qx(1) +Q(1)x− 〈x(3) +Qx(1) +Q(1)x, x〉

dx

= x(3) +Qx(1) +3

2Q(1)x. (4.27)

Now (4.27) is related to a null Lie quadratic V in so(3,R) or so(1, 2,R) as follows: we can

modify [51, Lemma 3], which gives a third order linear ODE for V , to work in sl(2,C)

using the identity (4.12):

Lemma 4.7. A null Lie quadratic V in so(3,R) or so(1, 2,R) satisfies

V (3) + ε(FV (1) − 1

2F (1)V ) = 0 (4.28)

where F = 〈V, V 〉 is a quadratic function of t, and ε = ±1 according to whether we are in

so(3,R) or so(1, 2,R).

Proof. Differentiating (4.5) and substituting (4.5) gives

V (3) = [[V (1), V ], V ].

Using (4.13) gives

V (3) = −ε(〈V, V 〉V (1) − 〈V (1), V 〉V )

where, by [51, Corollary 1], we have 〈V, V 〉 is a quadratic in t.

The coefficients of F depend upon the initial conditions of V . Differentiating (4.28) twice

and letting Z = V (2), we have

Z(3) + ε(FZ(1) +3

2F (1)Z) = 0. (4.29)

42

Thus Z and x satisfy the same differential equation if F = −Q. Since this is a third order

linear differential equation, if Z(t) : t ∈ C spans sl(2,C) then the general solution of

(4.27) can be found as a linear transformation of Z. In fact we can say when this fails.

Proposition 4.8. V (t) : t ∈ C fails to span sl(2,C) if and only if V arises from the

degenerate form of Proposition 4.4.

Proof. If V can be written in the degenerate form given, then immediately V is contained

in the subspace of upper triangular elements of sl(2,C).

Now suppose that V (t) does not span sl(2,C). Then V (0), V (1)(0), V (2)(0) are lin-

early dependent. Then either: [V (1)(0), V (0)] = V (2)(0) lies in the span of V (0), V (1)(0);

or, V (0), V (1)(0) are linearly dependent. In either case, V (0), V (1)(0) generate a soluble

subalgebra of sl(2,C), and this is the condition that implies that V can arises from the

degenerate form of Proposition 4.4.

Now we can compute the JK-quadratics in the sphere S2 using Proposition 4.6 and

Equations (4.24) and (4.21).

Proposition 4.9. The nondegenerate JK-quadratics in S2 ⊆ E3 are of the form

MψE3

(1

c

(−2a(1)U2 + v(1)U1 + (2a(1)v + v(2))U0

)).

with U0, U1, U2 defined as in Section 4.5, where a, b, c satisfy the conditions of Proposition

4.6, and M is a similarity of R3 with scaling factor chosen so that the curve lies in S2.

The nondegenerate JK-quadratics in H2 ⊆ L3 are of the same form, with ψL3 substituted

for ψE3, where a, b, c are real.

Note that this also gives the JK-quadratics in Sn for n > 2, since any given initial

conditions in Sn (as vectors in Rn+1) are contained in some 3-dimensional subspace and

the JK-quadratics can be solved in the sphere given by intersecting this 3-space with Sn.

4.9 Acknowledgments

I thank Professor Lyle Noakes for his very helpful advice and also for drawing my attention

to the link between null Lie quadratics and Euler spirals. I am grateful to the reviewer

for informing me of Reference [21].

43

44

Chapter 5

Jupp and Kent’s cubics in Lie groups1

Abstract

We study a generalisation of the cubic polynomial to Riemannian manifolds

and other affine connection spaces, modifying the differential equation x(4) = 0

by replacing higher derivatives with covariant derivatives. In matrix groups,

with two particular choices of a left invariant connection, we can convert the

equation into a system of first order linear differential equations. We give

asymptotics in a generic case in GL(n) and in the n-sphere.

5.1 Introduction

Let M be an affine connection space, i.e. a smooth manifold with a connection ∇ on the

tangent bundle. For a smooth curve x : R→M , and for any vector field X defined along

x, the connection gives us a time derivative ∇x(1)X. As a shorthand we write ∇tX; it is

also elsewhere written ∇d/dtX or DXdt .

The purpose of this paper is to investigate the equation

∇3tx

(1) = 0, (5.1)

in a Lie group, and in particular in GL(n). For the interpolation and approximation of

time series data on a sphere, [38] proposed the use of curves satisfying (piecewise) (5.1),

where ∇ is the Levi-Civita connection on the sphere. This is an analogue of the equation

x(4) = 0 of a cubic in Euclidean space. We are calling the solutions of (5.1) JK-cubics

after the inventors of the equation, Jupp and Kent.

As an alternative, Jupp and Kent also suggested using curves that are critical points

of ∫ T

0〈∇tx(1),∇tx(1)〉dt (5.2)

They found the Euler-Lagrange equation in the sphere but found that solving it was

not feasible. The variational problem (5.2) was independently proposed by [24, 57] for

1To appear as [63]. A couple of wording changes have been made and a paragraph on left invariant

connections has been shortened to avoid duplicating material in Section 1.8.

45

Figure 5.1: An example of a JK-cubic in a 2-sphere (left) and a JK-quadratic (right).

interpolation in arbitrary Riemannian manifolds. In general the Euler-Lagrange equation

is

∇3tx

(1) +R(∇tx(1), x(1))x(1) = 0 (5.3)

where R is the Riemannian curvature tensor. Solutions are known as Riemannian cubics.

Riemannian cubics are perhaps a more natural choice of interpolant curves than JK-

cubics, because they can be described as minimal. However there are some reasons to use

JK-cubics:

• JK-cubics are more predictable. In this paper we will give some asymptotic formulae

for generic JK-cubics in GL(n) and the sphere Sn for any n. The asymptotics of

Riemannian cubics require much more ingenuity, even in the case of SO(3), as seen

in [52];

• For higher order splines, the integral (5.2) can be replaced by∫ T

0〈∇k−1t x(1),∇k−1t x(1)〉dt. (5.4)

This gives a generalisation of odd order polynomials, suggested in [13] to produce

geometric splines of class C2k−2. They are also studied in [42, 66] and referred to as

Curves of class Dk−1 or higher order geodesics. But in general, geodesics of order k

are not geodesics of order k+ 1 [42, Section 5.2]. Equation (5.1) can be modified to

∇kt x(1) = 0 (5.5)

to generalise polynomials of any order, and solutions of ∇kt x(1) = 0 are automatically

solutions of ∇k+1t x(1) = 0. A version of Jacobi fields for solutions of (5.5) are studied

in [46]. (Motivated by the theory of parabolic splines, Chapter 4 of the present thesis

studies solutions of ∇2tx

(1) = 0, called JK-quadratics. In certain Lie groups and in

spheres, generic JK-quadratics can be solved in terms of special functions. Examples

of a JK-cubic and a JK-quadratic are given in Figure 5.1.) In [46], solutions of (5.5)

are called (k+1) geodesics, with k = 1 corresponding to geodesics, and critical points

of (5.4) are in [66] called k-geodesics, with k = 1 corresponding to geodesics.

46

Figure 5.2: A JK-cubic (darker curve) and a Riemannian cubic (lighter curve) in the

sphere, with boundary conditions x(0) = (1, 0, 0)T , x(1)(0) = (0, 1, 0)T and x(1) =

(0.6, 0, 0.8)T , x(1)(1) = (−4,−6, 3)T .

Figure 5.3: The JK-cubic and Riemannian cubic of Figure 5.2, extended for further

time.

• Equation (5.1) makes sense in the more general situation of an affine connection

space. Although Equation (5.3) has meaning in an affine connection space, the

variational problem (5.2) does not. Therefore in an arbitrary affine connection space,

Riemannian cubics are no more natural than JK-cubics.

Figure 5.2 compares a JK-cubic with example boundary conditions, with a Riemannian

cubic with the same boundary conditions. Figure 5.3 shows extended solutions of the same

ODEs.

We are studying Equation (5.1) in a Lie group. The necessary theory of left-invariant

connections on a Lie group can be found in [32, Section II.1.3] or [71, Section 6] and a

summary can be found in Section 1.8 of the present thesis.

The solution of (5.1) with either the (-)- or the (0)-connection could be used to interpo-

late time series data in a Lie group; for example a sequence of points in the rotation group

SO(3). (By considering the opposite group, we see that results on the (+)-connection

follow from those on the (-)-connection.)

47

When the Lie group has a faithful representation in GL(n), using the (-)-connection,

(5.1) can be reduced to a linear system of n first order differential equations:

x(1) = (A0 +A1t+A2t2)x (5.6)

More surprisingly, in the (0)-connection, (5.1) can be reduced to a pair of linear systems

of n first order differential equations. We give asymptotic formulae for both of these

connections. We also show how JK-cubics in spheres Sn can be found from those in

SU(2).

5.2 JK-Cubics in Lie groups

Let G be a Lie group with Lie algebra g, and a left invariant connection ∇. We treat g as

the tangent space to G at the identity. Let Lx(y) = xy. For a curve x : R → G and any

vector field X defined along x, let X(t) = (dLx(t)−1)X(t). Then X(t) is a curve in the Lie

algebra. In particular let V = x(1). the curve V in g is called the left Lie reduction of x.

Then for a vector field X defined along x we have

∇tX(t) = X(1)(t) + α(V (t), X(t))

where α : g × g → g is the map defined from ∇ as in Section 5.1. It follows that for the

(-)-connection,

∇tX(t) = X(1) (5.7)

and for the (0)-connection,

∇tX(t) = X(1) +1

2[V,X]. (5.8)

5.2.1 (-)-connection

In this case it follows from (5.1), (5.7) that V satisfies the differential equation V (3) = 0

and so

V = A0 +A1t+A2t2

for some constants A0, A1, A2 in g. Thus

x(1) = dLx(A0 +A1t+A2t2). (5.9)

If G is a matrix group, this becomes x(1) = x(A0 + A1t + A2t2). Transposing gives

x(1)T = (AT0 + AT1 t + AT2 t2)xT , which is of the form (5.6). Asymptotic solutions of this

equation are found under generic conditions in Section 5.3.

5.2.2 (0)-connection

In [51] it is found useful in the study of Riemannian cubics to find a differential equation

for V = dLx−1x(1). We find this useful to help visualise properties of solutions of (5.1) in

48

-20 0 20

-10

0

10

20

-8

-6

-4

-2

0

Figure 5.4: An example of the left Lie reduction of a JK-cubic in SO(3).

SO(3). Following [51] in applying (5.8) gives

dLx−1∇tx(1) = V (1)

dLx−1∇2tx

(1) = V (2) +1

2[V, V (1)]

dLx−1∇3tx

(1) = V (3) + [V, V (2)] +1

4[V, [V, V (1)]]

Thus ∇3tx

(1) = 0 gives a third order differential equation for V :

V (3) + [V, V (2)] +1

4[V, [V, V (1)]] = 0. (5.10)

An example is given in Figure 5.4 when G is SO(3). There, the Lie algebra is so(3) which

is isomorphic to E3 which is R3 with the cross product.

5.3 Asymptotics of the linear system under generic conditions

Let y be a curve in the space Cn×n of complex, n× n matrices, and let y0, y1, y2, · · · be a

sequence of constant complex n× n matrices. We write

y(t) ∼ y0 + y1t−1 + y2t

−2 + · · ·

and say that y0 + y1t−1 + y2t

−2 + · · · is the asymptotic expansion of y as t → ∞, if, for

every i,

y(t)− (y0 + y1t−1 + · · ·+ yit

−i) = o(t−i) as t→∞.

This does not require the series y0 + y1t−1 + y2t

−1 + · · · to converge anywhere. We also

write

y(t) ∼ (y0 + y1t−1 + y2t

−2 + · · · )z(t)

if y(t)z(t)−1 ∼ (y0 + y1t−1 + y2t

−2 + · · · ). Suppose A(t) has an asymptotic expansion

A ∼ A0 + A1t−1 + A2t

−2 + · · · , and that A0 has distinct eigenvalues, and let q be a non-

negative integer. Then ([17, Chapter 5] or [84, Chapter IV]) there is a solution y(t) of the

linear system

y(1)(t) = tqA(t)y(t) (5.11)

49

with asymptotic expansion

y(t) ∼ (y0 + y1t−1 + y2t

−2 + · · · )tD exp(Qq+1tq+1 + · · ·+Q1t), (5.12)

where y0, y1, · · · are constant matrices, and D,Qq+1, · · · , Q1 are constant diagonal matri-

ces. (Here tD means exp(D ln(t)). Now, D is diagonal; suppose its entries are d1, · · · , dn.

Then tD is diagonal with entries td1 , · · · , tdn .) An asymptotic expansion for an arbitrary

solution of (5.11) is given by multiplying (5.12) on the right by an arbitrary constant

matrix yR. (Although it is not so simple to find the yR corresponding to the solution of

(5.11) with, say, y(0) = 1.)

We apply this to Equation (5.6).

Theorem 5.1. Suppose A2 has distinct eigenvalues. Any solution x : R→ Cn×n of (5.6)

has asymptotic expansion

x(t) ∼ (x0 + x1t−1 + x2t

−2 + · · · )tD exp(Q3t

3 +Q2t2 +Q1t

)xR (5.13)

where xR, x0, x1, · · · are constant complex matrices, and D,Q3, Q2, Q1 are diagonal con-

stant matrices.

Proof. Write (5.6) as

x(1)(t) = t2A(t)x(t) (5.14)

with A(t) = A2 +A1t−1 +A0t

−2. This is now in the form (5.11).

As explained in Section 5.2.1, we now have an asymptotic expansion for (the transpose

of) almost all JK-cubics in GL(n) with the (-)-connection.

5.4 (0)-connection: Conversion to two systems of linear ODEs

Suppose our Lie group G is actually a matrix group, and let ∇ be the (0)-connection of

Equation (5.8). Let x : R → G be a smooth curve and let X be a vector field defined

along x. We know from Equation (5.8) that

x−1(∇tX) =d

dt(x−1X) +

1

2[x−1x(1), x−1X]

thus it follows that

∇tX = X(1) − 1

2

(x(1)x−1X +Xx−1x(1)

).

Now, for any X : R→ Cn×n, (not just those tangent to G at x) define

M : X 7→ X(1) − 1

2

(x(1)x−1X +Xx−1x(1)

). (5.15)

Lemma 5.2. For any X,Y : R→ Cn×n, the operator M satisfies the product-like formulae

M(Xx−1Y

)= (MX)x−1Y +Xx−1(MY ), (5.16)

M (fX) = f (1)X + fMX. (5.17)

50

Proof. (Equation (5.16):)

M(Xx−1Y ) =d

dt

(Xx−1Y

)− 1

2

(x(1)x−1(Xx−1Y ) + (Xx−1Y )x−1x(1)

)= X(1)x−1Y −Xx−1x(1)x−1Y +Xx−1Y (1)

− 1

2

(x(1)x−1(Xx−1Y ) + (Xx−1Y )x−1x(1)

)=

(X(1) − 1

2

(x(1)x−1X +Xx−1x(1)

))x−1Y

+Xx−1(Y (1) − 1

2

(x(1)x−1Y + Y x−1x(1)

))= (MX)x−1Y +Xx−1(MY ).

(Equation (5.17):)

M(fX) =d

dt(fX)− 1

2

(x(1)x−1(fX) + (fX)x−1x(1)

)= f (1)X + fMX.

We will say that a curve Z : R → Cn×n is an M -constraint on x if MZ = 0. Compare

with the useful Lax constraints of [54].

Corollary 5.3. If X and Y are M -constraints then Xx−1Y is an M -constraint. A lin-

ear combination (over C) of M -constraints is also an M -constraint, and the set of M -

constraints form a complex vector space of dimension n2.

Proof. If MX = MY = 0 then by (5.16), we have M(Xx−1Y ) = (MX)x−1Y +

Xx−1(MY ) = 0. A linear combination of M -constraints is an M -constraint by the linear-

ity of M . Since it is a first order operator, there is a unique M -constraint Z with any given

initial condition Z(t0). Thus the set of M -constraints forms a vector space of dimension

n2.

Suppose that x is a solution of

∇3tx

(1) = 0.

Without loss of generality we may assume that x(0) = 1 since the connection is left

invariant. Write Eij for the matrix with a 1 in the (i, j) component and a 0 everywhere

else. Let Zij be the M -constraint with Zij(0) = Eij .

Proposition 5.4.

x(1) =∑

f ijZij (5.18)

where each f ij is a quadratic function of t.

Proof. Since Zij(t) form a basis of Cn×n for every t, we can write x(1) =∑f ijZij for

some functions f ij . Repeatedly applying ∇t, using the fact that ∇tX = MX and using

(5.17), we have

∇tx(1) =∑

(f ij)(1)Zij ,

∇2tx

(1) =∑

(f ij)(2)Zij ,

0 = ∇3tx

(1) =∑

(f ij)(3)Zij ,

51

and using the fact that Zij(t) form a basis, it follows that (f ij)(3) = 0, and f ij is a

quadratic for each i, j.

Proposition 5.5. Let σ : R→ Cn×n be a solution of

σ(1) − 1

2x(1)x−1σ = 0. (5.19)

Then Zx−1σ is also a solution of (5.19) for any M -constraint Z.

Similarly, if ρ is a solution of

ρ(1) − 1

2ρx−1x(1) = 0 (5.20)

then ρx−1Z is a solution of (5.20) for any M -constraint Z.

Proof. Calculate

d

dt(Zx−1σ)− 1

2x(1)x−1(Zx−1σ)

= Z(1)x−1σ − Zx−1x(1)x−1σ + Zx−1σ(1) − 1

2x(1)x−1Zx−1σ

=

(Z(1) − 1

2(Zx−1x(1) + x(1)x−1Z)

)x−1σ + Zx−1

(σ(1) − 1

2x(1)x−1σ

)= 0

and similarly for ρ.

Theorem 5.6. Let F = 12

∑f ijEij. Then x can be solved as

x(t) = σ0(t)ρ0(t) (5.21)

where σ0, ρ0 : R→ Cn×n are solutions of the first order linear systems

σ(1)0 = σ0F, (5.22)

ρ(1)0 = Fρ0, (5.23)

with initial conditions σ0(0) = ρ0(0) = 1.

Proof. Equation (5.19) is a first order linear system of equations for each column of σ,

and it follows that if σ0 is a solution whose columns are all independent, then any other

solution σ can be found as σ(t) = σ0(t)A for some constant matrix A. We shall take σ0 to

be the solution of (5.19) with σ0(0) = 1. Then, because Zijx−1σ0 is a solution of (5.19)

and

Zij(0)x−1(0)σ0(0) = Eij

it follows that

Zijx−1σ0 = σ0Eij

and so

σ(1)0 =

1

2x(1)x−1σ0 =

1

2

(∑f ijZijx

−1)σ0 =

1

2σ0

(∑f ijEij

).

Similarly, letting ρ0 be the solution of (5.20) with ρ0(0) = 1, we have

ρ(1)0 =

1

2

(∑f ijEij

)ρ0.

Lastly, we compute from (5.15) thatMx = 0, and from (5.15),(5.19),(5.20) thatM (σ0ρ0) =

0, and since M is a first order linear differential operator, and x(0) = 1 = σ0(0)ρ0(0), it

follows that x = σ0ρ0.

52

In the following Corollary we will say that

x(t) ∼ z1(t)(x0 + x1t−1 + x2t

−2 + · · · )z2(t)

to mean that

z1(t)−1x(t)z2(t)

−1 ∼ x0 + x1t−1 + x2t

−2 + · · ·

as in Section 5.3.

Corollary 5.7. Suppose a solution x of (5.1) satisfies the generic requirement that

∇2tx

(1)(0)

has distinct eigenvalues. Then there exist constant matrices xL, xR, xi, (i = 0, 1, 2, · · · )and constant diagonal matrices DL, DR, QL,i, QR,i, (i = 1, 2, 3) such that

x(t) ∼ xL exp(QL,3t

3 +QL,2t2 +QL,1t

)tDL

· (x0 + x1t−1 + x2t

−2 + · · · )tDR exp(QR,3t

3 +QR,2t2 +QR,1t

)xR (5.24)

Proof. We know that x(1) =∑f ijZij where each of the f ij are quadratic functions of

t. Since M -constraints form a vector space, we can write x(1) = Z0 + Z1t + Z2t2 for

M -constraints Z0, Z1, Z2, and Equations (5.22),(5.23) become

σ(1) = σ(E0 + E1t+ E2t2) (5.25)

ρ(1) = (E0 + E1t+ E2t2)ρ (5.26)

where we can find E0, E1, E2 from the initial conditions of x:

E0 = Z0(0) = (Z0 + Z1t+ Z2t2)|t=0 = x(1)(0) (5.27)

E1 = Z1(0) = (Z1 + 2Z2t)|t=0 = ∇tx(1)(0) (5.28)

E2 = Z2(0) =1

2(2Z2)|t=0 =

1

2∇2tx

(1)(0). (5.29)

Generically, ∇2tx

(1)(0) has distinct eigenvalues.

Applying Theorem 5.1 to Equation (5.25) (actually to its transpose) and to (5.26) gives

σ(t) ∼ xL exp(QL,3t

3 +QL,2t2 +QL,1t

)tDL(σ0 + σ1t

−1 + σ2t−2 + · · · )

ρ(t) ∼ (ρ0 + ρ1t−1 + ρ2t

−2 + · · · )tDR exp(QR,3t

3 +QR,2t2 +QR,1t

)xR,

where xL, xR, σi, ρi are constant matrices and DL, DR, QL,i, QR,i are constant diagonal

matrices. Now Equation (5.21) gives (5.24), since the product of the two series σ0 +

σ1t−1 + σ2t

−2 + · · · and ρ0 + ρ1t−1 + ρ2t

−2 + · · · is a series x0 + x1t−1 + x2t

−2 + · · · .

5.5 Spheres

Here we consider the JK-cubics in Sn with the Levi-Civita connection corresponding to

the metric inherited from Rn+1. An example of a JK-cubic is shown in Figure 5.1.

Suppose n > 3. A solution of ∇3tx

(1) = 0 is defined by its inital conditions at some

time, x(t0), x(1)(t0),∇tx(1)(t0),∇2

tx(1)(t0). These initial conditions are contained in some

53

4-dimensional subspace of Rn+1, and the intersection of this subspace with Sn is a 3-sphere

which is a totally geodesic submanifold of Sn−1. This 3-sphere is isometric to the usual

3-sphere S3. Thus the JK-cubics in any sphere Sn can be found from those in S3 (If n < 3

then we may consider Sn as a totally geodesic submanifold of S3). Now, the restriction of

the map x

y

z

w

→(x+ iy z + iw

−z + iw x− iy

)

to S3 is a diffeomorphism from S3 to the Lie group SU(2), and it takes the Riemannian

metric on S3 to a bi-invariant inner product on SU(2). The Levi-Civita connection for

this inner product is bi-invariant and torsion-free, so it is the (0)-connection. Therefore

the JK-cubics in S3 can be found from those in SU(2).

Elements of su(2) generically have distinct eigenvalues. Therefore Corollary 5.7 gives

asymptotics for JK-cubics in spheres.

5.6 Acknowledgments

My thanks go to Professor Lyle Noakes for many helpful discussions and his encouragement

and advice.

54

Chapter 6

Lie algebras and asymptotic series for

solutions of linear ordinary differential

equations

6.1 Introduction

In Chapter 5 we investigated the asymptotics of JK-cubics in a Lie group G of n× n real

matrices, with either the (-)-connection or the (0)-connection. With the (-)-connection,

we reduced the equation for a JK-cubic to an n × n linear differential equation y(1)(t) =

(A2t2+A1t+A0)y(t), where the Ai are n×n real matrices in the Lie algebra g of G. With

the (0)-connection, we got a pair of such equations. These linear systems are a special

case of

y(1)(t) = A(t)y(t), (6.1)

A(t) =∑

−∞<i≤qAit

i,

where q is a non-negative integer and the sum for A(t) is convergent. When Aq has distinct

eigenvalues, it is easy to produce a formal solution Y of (6.1) of the form

Y =

∑−∞<i≤0

Yiti

tD exp(Qq+1tq+1 + · · ·+Q0), (6.2)

for Yi ∈ Cn×n and D,Qi diagonal.

From [17, Chapter 5] or [84, Chapter IV], if Y is a formal solution of the form (6.2), then

there is an actual solution y such that Y is the asymptotic expansion of y, namely, for any

i < 0,

y(t) exp(−Qq+1tq+1 − · · · −Q0)t

−D −∑i≤k<0

Yktk = o(ti) as t→∞.

If G = GL(n) then the set of Aq ∈ g with distinct eigenvalues is open and dense. We used

this in Theorem 5.1 and the following discussion to describe the asymptotics for generic

JK-cubics in GL(n) with the (-)-connection. In Corollary 5.7 we described the asymptotics

55

of generic JK-cubics in GL(n) with the (0)-connection. However, in some Lie algebras it

is possible that no elements have distinct eigenvalues. For example, suppose G = SE(3)

is the group of orientation-preserving isometries of R3. It is common to write elements of

SE(3) as 4 × 4 matrices

(M v

0 1

)where M ∈ SO(3) and v is a 3 × 1 column vector (see

for example [11, Section 5.1] or [48, Section 14.7]). An element of the Lie algebra se(3)

of SE(3) is then written as a matrix of the form

(Ω v1

0 0

)where Ω is a skew-symmetric

3 × 3 matrix and v1 is a 3 × 1 column vector. Written this way, any element of se(3)

has zero as a repeated eigenvalue. Thus the theory of [17, Chapter 5] or [84, Chapter IV]

does not apply to Equation (6.1). More complicated asymptotic solutions exist but the

computation becomes more difficult and involves decisions that depend on the particular

nature of the coefficients Ai [84, Chapter V]. We will find in this chapter that regardless

of g, there is still a generic condition on the leading coefficient Aq for which asymptotic

expansions for solutions can be found easily.

When Aq is a regular element of the Lie algebra g, there is a Cartan subalgebra

h = Y ∈ g : (adAq)sY = 0 for some s ≥ 1 . (6.3)

(See section 1.7.) For some power series in g, we can define their exponentials, and we will

show that there is a formal solution of (6.1) of the form

Y = exp

∑−∞<i≤−1

Biti

exp

∑−∞<i≤w

Citi

tD (6.4)

where w ∈ Z, all Bi ∈ g, and all Ci, D ∈ h.

We also show that (after complexification, if necessary) the expression (6.4) can be con-

verted into

Y =

∑−∞<i<iY ,0≤j≤jY

Yi,jti ln(t)j

tR exp (Qwtw + · · ·+Q0) (6.5)

(iY ∈ Z, jY ∈ Z, jY ≥ 0, Yi,j , R,Qi ∈ Cn×n, all R,Qi simultaneously diagonalisable).

Once a basis has been chosen in which R,Qi are all diagonal, the series (6.5) is a special

case of the asymptotic series considered in [17, Section 5.6]. (The general case allows frac-

tional powers of t.) It follows that there exist solutions of (6.1) which are asymptotically

described by the formal solution (6.4).

The advantages of (6.4) over (6.2) are as follows.

• The calculation of the terms Bi, Ci, D is purely Lie algebraic, in the following sense.

If g and g′ are matrix Lie algebras and φ : g→ g′ is an isomorphism, then a formal

solution of

y(1)(t) = φ(A(t))y(t)

is

Y = exp(φB) exp(φC)tφ(D)

where φB =∑

−∞<i≤−1φ(Bi)t

i and φC =∑

−∞<i≤wφ(Ci)t

i.

56

Therefore for a given (abstract) Lie algebra, the formal solution (6.4) does not depend

on the choice of representation.

• It is possible that no elements of g have distinct eigenvalues. However the condition

that Aq is regular is generic for elements Aq of g, regardless of the structure of g.

To say that (6.4) is a formal solution we must be able to differentiate it. We can differ-

entiate (6.4) using the product rule, together with ddt t

D = (Dt−1)tD, and the rule for the

derivative of exp,

d

dtexp(X) =

(1− exp(−adX)

adX

)X(1), (6.6)

for a series X. The following theorem summarises the results of this chapter.

Theorem 6.1. 1 Let g be a real or complex matrix Lie algebra, and let A be a formal

series∑−∞<i≤q Ait

i with each Ai ∈ g. Suppose that Aq is a regular element of g. Let

h be the Cartan subalgebra defined by Equation (6.3). It is possible to choose w ∈ Z and

Ci, Bi, D such that Y given by (6.4) is a formal solution of Y (1) = AY . It can be converted

to an asymptotic solution of (6.1) of the form (6.5). 2 Each Yi,j and Qi depends only

on finitely many terms of B,C. If∑−∞<i≤q Ait

i converges to a function A(t) for all

sufficiently large t then there is a solution y of (6.1) with the formal solution (6.4) as its

asymptotic expansion. The general solution of (6.1) is found by multiplying y on the right

by an arbitrary constant matrix.

Proof. Existence of a formal solution of the form (6.4) follows from Lemmas 6.5 and 6.6

below. Conversion to the form of [17] follows from Lemmas 6.2, 6.3. Lemma 6.4 gives the

existence of an actual solution y with (6.4) as its asymptotic expansion.

6.2 Application to null Riemannian cubics

Let G be a (real) matrix group with the (0)-connection and let g be its Lie algebra. Let

x : R→ G be a Riemannian cubic, i.e. a solution of (2.5). As in Section 2.3, the left Lie

reduction V = x−1x(1) is a Lie quadratic, a solution of

V (2) = [V (1), V ] + C (6.7)

where C is a constant. The solutions with C = 0 are called null Lie quadratics. Corre-

sponding Riemannian cubics are called null . Solutions of

V (2)(t) = 0 (6.8)

are also solutions of (6.7) and they are called affine Lie quadratics. Corresponding Rie-

mannian cubics are affine. It was shown in [53] that when x is a null cubic, x1 : t 7→ x(t)−1

is affine. Asymptotics have been found to high accuracy in the special case of SO(3) [56].

1I recently became aware of [7]. The theory there would possibly simplify the present chapter.2In the real case, complexification is necessary to make R,Qi simultaneously diagonalisable.

57

The solutions of (6.8) are V = V1t+ V0. Therefore affine cubics are solutions of

x(1)1 (t) = x1(t)(V1t+ V0).

and corresponding null cubics are solutions of

x(1)(t) = −(V1t+ V0)x(t). (6.9)

If V1 is a regular element of g, there is a formal asymptotic solution of the form (6.4).

6.3 Application to JK-cubics

Let G be a (real) matrix Lie group with Lie algebra g. Let x : R→ G be a JK-cubic, i.e.

a solution of Equation (2.4). Thanks to Theorem 5.6, we can write

x(t) = σ0(t)ρ0(t)

where σ0 : R→ Cn×n is a solution of

σ(1)0 (t) = σ0(F2t

2 + F1t+ F0)

and ρ0 : R→ Cn×n is a solution of

ρ(1)0 (t) = (F2t

2 + F1t+ F0)ρ0(t)

for constant matrices F0, F1, F2. By Equations (5.27), (5.28), (5.29), the matrices F0, F1, F2

are all in the Lie algebra g. Theorem 6.1 shows that if F2 is regular then σ0, ρ0 can be

described asymptotically by

σ−10 ∼ exp(B−1,Lt−1 +B−2,Lt

−2 + · · · ) exp(Cw,Ltw + Cw−1,Lt

w−1 + · · · )tDLxLρ0 ∼ exp(B−1,Rt

−1 +B−2,Rt−2 + · · · ) exp(Cw,Rt

w + Cw−1,Rtw + · · · )tDRxR

where Bi,L, Bi,R are constant elements of g, Ci,L, Ci,R, DL, DR are constant elements of

some Cartan subalgebra of g, and xL, xR are constant matrices. Allowing for an abuse of

notation we write

x(t) ∼(exp(B−1,Lt

−1 +B−2,Lt2 + · · · ) exp(Cw,Lt

w + Cw−1,Ltw−1 + · · · )tDLxL

)−1· exp(B−1,Rt

−1 +B−2,Rt−2 + · · · ) exp(Cw,Rt

w + Cw−1,Rtw−1 + · · · )tDRxR.

6.4 Formal series and their exponentials

A special case of (6.5) is when the formal series is just a power series, i.e.

Y =∑

−∞<i≤iY

Yiti (6.10)

iY ∈ Z, Yi ∈ Cn×n.

The purpose of this section is to define, in specific situations, the exponential of a power

series X of the form (6.10). If Xi = 0 for all i ≥ 0, or if all Xi lie in a Lie algebra consisting

58

only of nilpotent matrices, then exp(X) will itself be of the form (6.10) (Lemma 6.2). If

all Xi lie in some nilpotent matrix Lie algebra h, then exp(X) will be of the form ∑−∞<i≤iY

Yiti

exp(Qwtw + · · ·+Q0), (6.11)

so that for D ∈ h, we can write exp(X)tD in the form (6.5) (Lemma 6.3). Most of this

section is dedicated to showing that differentiation and multiplication of exponentials work

the way we expect. We can then use this in Sections 6.5 and 6.6 to show the existence of

formal solutions.

If two series X,Y are of the form (6.5) with the same R,Qi, then they can be added.

The series Y can also be formally differentiated: its derivative is of the form (6.5) with

the same R,Qi. Series of the form (6.5) with R = Qi = 0 can be multiplied together, and

this multiplication is associative. Series of the form (6.10) can be added and multiplied.

If X is of the form (6.10) and Y is of the form (6.5) then XY is of the form (6.5) and

d

dt(XY ) = X(1)Y +XY (1). (6.12)

The definitions (6.5) and (6.10) can be extended to any algebra of endomorphisms of a

finite dimensional vector space. If g is a Lie subalgebra of Cn×n, write End(g) for the

algebra of all vector space endomorphisms of g. Suppose X is a series in End(g) of the

form (6.10), and Y is a series in g of either of the forms (6.5) or (6.10). Then the product

XY is a series of the form (6.5) or (6.10) respectively, and (6.12) holds. For a series

Y =∑−∞<i≤iY Yit

i in g we can define the series adY =∑−∞<i≤iY (adYi)t

i in End(g).

For a series X of the form (6.10) and a given integer s, write X(s) for the finite series

formed by deleting all terms Xktk for k < s. Then X(s) is a map defined at least on (0,∞).

If X is a series of the form (6.10), we can in certain situations define exp(X) to be a

series of the form (6.10), as follows. For i > iX let Xi := 0. Define, for each i, j ∈ Z, j ≥ 0,

Xi,j =∑

i1,··· ,ij∈Z,i1+···+ij=i

(Xi1Xi2 · · ·Xij

)(6.13)

(taking the empty sum to be zero and the empty product to be the identity). If X is a

finite series such that either (i) Xi = 0 for all i ≥ 0, or (ii) all Xi lie in some Lie algebra

of nilpotent matrices, then we can compute that

exp(X) =∑j≥0

1

j!Xj =

∑i

∑j

1

j!Xi,j

ti.

We use this formula to extend the definition of exp to infinite series satisfying (i) or (ii).

Lemma 6.2. Let X =∑−∞<i≤iX Xit

i be a series of the form (6.10) in a Lie subalgebra

g of Cn×n. Suppose either (i) Xi = 0 for i ≥ 0, or (ii) the Lie subalgebra g consists only

of nilpotent matrices. Define

exp(X) :=∑j≥0

1

j!Xj =

∑i

∑j

1

j!Xi,j

ti, (6.14)

59

and (1− exp(−X)

X

):=∑j≥0

(−1)j

(j + 1)!Xj =

∑i

∑j

(−1)j

(j + 1)!Xi,j

ti. (6.15)

Then:

1. exp(X) and 1−exp(−X)X are series of the form (6.10) in g;

2. If X satisfies (i) or (ii) then so does the series adX in End(g) so that 1−exp(−adX)adX

is defined: this satisfies the rule for the derivative of the exponential,

d

dt(exp(X)) = exp(X)

((1− exp(−adX)

adX

)X(1)

); (6.16)

3. For another series A =∑−∞<i≤iA Ait

i we have

exp(−X)A exp(X) = exp(−adX)A, (6.17)

and

exp(X) exp(−X) = 1; (6.18)

4. In case (i), the coefficient of ti in exp(X) is zero for i > 0. In case (ii), the coefficient

of ti in exp(X) is zero for i > (n − 1)iX . There is an integer v ≥ 0 such that the

coefficient Yi of each ti in the series exp(X) depends only on Xk for k ≥ i − v. In

case (i) we have v = 0, and in case (ii) we have v = max(0, (n− 2)iX);

5. For any integer k ≤ 0 we have

det

∑k≤i≤0

Yiti

= 1 + o(1). (6.19)

Proof. Properties 1 and 4: In both cases (i) and (ii), we see that each of the terms Xi,j

or adXi,j is a finite sum depending only on Xk for k ≥ i− v, and for all sufficiently large

i, we have Xi,j = adXi,j = 0 for all j. (Case (i): Let v = 0. Every nonpositive integer

has only finitely many partitions into negative integers, and any positive integer has no

partitions into negative numbers. Case (ii): Let v = max((n− 2)iX , 0). By one version of

Engel’s theorem (see [75, Theorem 11.10 and Corollary 11.11]), there is a basis in which

all Xi are strictly upper triangular, therefore all products of length greater than n− 1 are

zero. Any partition of an integer i < 0 into at most n − 1 integers bounded above by iX

has all its terms bounded below by i− (n− 2)iX .) Therefore Equations (6.14) and (6.15)

define series of the form (6.10).

Property 2: A proof of (6.16) (see, for example, [83, Theorem 2.14.3] or [82]) applies

to X(s). Given i, for all s ≤ i− v, the coefficients of ti in exp(X) and exp(X(s)) agree, and

the coefficients of ti in(1−exp(−adX)

adX

)and

(1−exp(−adX(s))

adX(s)

)agree. Therefore both sides of

(6.16) agree on the coefficient of ti. Since s can be chosen arbitrarily, we see that both

sides of (6.16) agree on the coefficient of every ti.

60

Property 3: Let Y = exp(−X)A exp(X) and

Z = exp(−X(s))A(s) exp(X(s)) = Ad(exp(−X(s)))A(s) = exp(−adX(s))A.

The coefficient of ti in Y depends only on finitely many terms of X and A so for a

sufficiently large choice of s (depending on i), the coefficients of ti in Y and Z agree. Since

s can be chosen arbitrarily, both sides of (6.17) agree on every term. A similar argument

with Z = exp(X(s)) exp(−X(s)) = 1 proves (6.18).

Property 5: Equation (6.19) holds because the constant term Y0 in exp(X) is 1, and

all other terms are o(1) as t→∞.

In a more general case than (ii) of Lemma 6.2, we can define the exponential of a series

of the form (6.10) to be a series of the form (6.5).

Lemma 6.3. Let X,Y be series of the form (6.10). Suppose that all Xi, Yi lie in some

nilpotent Lie subalgebra h of Cn×n. Then all adH, (H ∈ h) are nilpotent endomorphisms

of h, so that case (ii) of Lemma 6.2 implies exp(−adX) and(1−exp(−adX)

adX

)are series in

End(h) of the form (6.10). Suppose that D is some constant element of h. Then:

1. We can define exp(X)Y tD as a series of the form (6.5);

2. The definition satisfies (as we would compute using the product rule),

d

dt

(exp(X)Y tD

)= exp(X)

(((1− exp(−adX)

adX

)X(1)

)Y + Y (1) + Y Dt−1

)tD;

(6.20)

3. If A =∑−∞<i≤iA Ait

i is some other series in h then

exp(X)Y tD = A exp(X)tD ⇐⇒ Y = exp(−adX)A. (6.21)

and

Y exp(X)tD = A exp(X)tD ⇐⇒ Y = A. (6.22)

4. Suppose exp(X)Y tD is expanded in the form (6.5). There is an integer v ≥ 0 such

that for any i, j, the coefficients Yi,j depend only on Xk, Yk for k ≥ i − v. The

diagonal matrices Qi only depend on finitely many terms of X.

5. Suppose exp(X)tD is expanded in the form (6.5). For all sufficiently large negative

i,

det

∑i≤k≤iY ,0≤j≤jY

Yk,jtk ln(t)j

= 1 + o(1) as t→∞. (6.23)

Proof. Since h is a nilpotent Lie algebra, every adH, (H ∈ h) is a nilpotent endomorphism.

Therefore(1−exp(−adX)

adX

)is a series in End(h) of the form (6.10).

61

Property 1: From the representation theory of nilpotent Lie algebras ([83, Section

3.5]), we know that there is a basis of Cn such that all elements of h take the form

T1 0 0 · · · 0

0 T2 0 · · · 0

0 0 T3...

.... . .

0 0 Tp

where each Ti is a triangular block with only one eigenvalue. Write each Xi as Si +Ni for

Si diagonal and Ni strictly upper triangular in such a basis. Then Si and Ni commute.

Let

S =∑

−∞<i≤−1Sit

i Q =∑

0≤i≤wSit

i N =∑

−∞<i≤wNit

i.

Let SD and ND be the diagonal and strictly upper triangular parts of D. Define

exp(X)Y tD := exp(N) exp(S)Y exp(ND ln(t))tSD exp(Q) (6.24)

which is a series of the form (6.5) as follows: Since ND is strictly upper triangular, the com-

ponents of exp(ln(t)ND) are polynomial functions of ln(t). The factors exp(N) and exp(S)

are power series formed via Lemma 6.2. Thus multiplying exp(N) exp(S) exp(ln(t)ND)

gives a sum of the kind in (6.5). Lastly SD and Qi are diagonal.

Property 2: Equation (6.12) and cases (i),(ii) of Lemma 6.2 can be applied to Equation

(6.24) to show (6.20).

Property 3: We check (6.21). We have

exp(X)Y tD = exp(N) exp(S)Y exp(ND ln(t))tSD exp(Q)

A exp(X)tD = A exp(N) exp(S) exp(ND ln(t))tSD exp(Q)

so that equality implies

exp(N) exp(S)Y exp(ND ln(t)) = A exp(N) exp(S) exp(ND ln(t)).

Multiplying both sides on the right by exp(−ND ln(t)) gives

exp(N) exp(S)Y = A exp(N) exp(S),

equal as series of the form (6.10). Since all terms of S commute with all elements of g

we have exp(N)Y = A exp(N). For M ∈ Cn×n write adCn×nM for the image of M under

the adjoint representation of Cn×n. Applying (6.18), (6.17) gives Y = exp(−adCn×nN)A.

Noting that (adCn×nSi)H = 0 for any H ∈ h, we have Y = exp(−adCn×n(N +S+Q))A =

exp(−adCn×nX)A. Now, since the terms of X all lie in h, we have Y = exp(−adX)A,

where ad is the adjoint representation of h. The reverse implication is similar.

We check (6.22). Similarly to the argument for (6.21), we see that if Y exp(X)tD =

A exp(X)tD,

Y exp(N) exp(S) exp(ND ln(t)) = A exp(N) exp(S) exp(ND ln(t)).

62

Multiplying on the right by exp(ND ln(t)) gives Y exp(N) exp(S) = A exp(N) exp(S).

Multiplying on the right by exp(−S) exp(−N) and using (6.18) gives Y = A.

Property 4: The coefficient Yi,j in the expansion of exp(X)Y tD depends only on the

coefficient of ti in exp(N) exp(S)Y , which depends on products of the form Ni1,j1Si2,j2Yi3

for i1+ i2+ i3 = i. Now, by property 4 of Lemma 6.2, we have Ni1,j1 = 0 for i1 > (n−1)iX

while Si2,j2 = 0 for i2 > 0 and Yi3 = 0 for i3 > iY . Partitions i1 + i2 + i3 = i with

i1 ≤ (n − 1)iX , i2 ≤ 0, i3 ≤ iY must also satisfy i1 ≥ i − iY , i2 ≥ i − (n − 1)iX − iY , i3 ≥i− (n−1)iX . Let v = max(0, iY , (n−1)iX + iY , (n−1)iX). We also see that each Qi only

depends on Xk for k ≥ 0.

Property 5: Let i ∈ Z. Choose s < 0 so that, for all k ≥ i, the coefficient Yk,j depends

only on N`, S` for ` ≥ s. Then ∑i≤k≤iY ,0≤j≤jY

Yk,jtk ln(t)j

=(exp(N(s)) exp(S(s)) exp(ND ln(t))

)+ o(ti) as t→∞.

The determinant of an n×n matrix is a degree n polynomial function of the components.

Therefore if i is any sufficiently large negative integer (depending on n and iX),

det

∑i≤k≤iY ,0≤j≤jY

Yk,jtk ln(t)j

= det(exp(N(s)) exp(S(s)) exp(ND ln(t))

)+ o(1) as t→∞

= (1)(1 + o(1))(1) + o(1) as t→∞

= 1 + o(1) as t→∞,

using S(s) = o(1) and the fact that the exponential of a nilpotent matrix has determinant

1.

Lemma 6.4. Let Y be a formal solution of Y (1) = AY of the form (6.4). Expand Y into

the form (6.5) using Lemmas 6.2 and 6.3. Then there exists a solution y of (6.1) such

that for any integer i < 0,

y(t) exp(−Qwtw − · · · −Q0) =

∑i≤k≤iY ,0≤j≤jY

Yk,jtk ln(t)j

+ o(ti). (6.25)

The general solution of (6.1) is found by multiplying y on the right by an arbitrary constant

matrix.

Proof. It follows from the discussion in [17, Section 5.6] that there is a solution y of (6.1)

which satisfies (6.25). We prove that the matrix y(t) has full rank for some t, so that the

general solution of (6.1) is given by multiplying y on the right by an arbitrary constant

matrix. Using Equations (6.19), (6.23) we see that the expansion of Y in the form (6.5)

satisfies

det

∑i≤k≤iY ,0≤j≤jY

Yk,jtk ln(t)j

= 1 + o(1) as t→∞.

The determinant of an n × n matrix is a degree n polynomial function of its entries, so,

choosing i to be a sufficiently large negative number (depending on n and iY ), and taking

determinants of both sides of (6.25), we see det(y(t) exp(−Qwtw − · · · − Q0)) = 1 + o(1)

as t→∞. Therefore y has full rank for all sufficiently large t.

63

6.5 Formal solution: nilpotent Lie algebra

Lemma 6.5. Let g be a nilpotent real or complex matrix Lie algebra of dimension r, and

let A = Aqtq +Aq−1t

q−1 + · · · be an asymptotic series in g. For a constant D ∈ g and an

asymptotic series C ∼ Cwtw + Cw−1tw−1 + · · · , write

Y = exp(C)tD. (6.26)

Choose C0 ∈ g arbitrarily. It is possible to choose Ci, (i 6= 0) and D so that Y formally

satisfies

Y (1) = AY. (6.27)

If g′ is another matrix Lie algebra and φ : g → g′ is a Lie algebra isomorphism, then

Y1 = exp(φC)tφD is a formal solution of Y(1)1 = (φA)Y1.

Proof. From Lemma 6.3,

Y (1) = exp(C)

((1− exp(−adC)

adC

)C(1) +Dt−1

)tD.

Substituting into (6.27) and using (6.21), we see that (6.27) is equivalent to(1− exp(−adC)

adC

)C(1) +Dt−1 = exp(−adC)A. (6.28)

Equation (6.28) depends on the structure of g but not on its representation. Thus we

see that if φ : g → g′ is a Lie algebra isomorphism and Y is a solution of (6.27) then

Y1 = exp(φC)tφD is a solution of Y (1) = (φA)Y1.

Expanding the series(1−exp(−adC)−adC

)and exp(−adC), and using the fact that since g

is nilpotent and has dimension r, for any X ∈ g we have (adX)r = 0:(r−1∑i=0

1

(i+ 1)!(adC)i

)C(1) +Dt−1 =

(r−1∑i=0

1

i!(adC)i

)A. (6.29)

Isolating Dt−1 and the first term in the sum on the left of (6.29),

C(1) +Dt−1 =

(r−1∑i=0

(−1)i

i!(adC)i

)A−

(r−1∑i=1

1

(i+ 1)!(adC)i

)C(1). (6.30)

Choose a basis U0, · · · , Ur−1 of h, of the form in Chapter 3. Thus [Uk, U`] is contained in

the span of the elements Umax(k,`)+1, · · · , Ur−1. For an element X of g, define πk(X) to be

the k-coordinate of X written in the basis U0, · · · , Ur−1, so that∑

0≤k≤r−1 πk(X)Uk = X.

Extend πk to power series by πk∑−∞<i≤iX Xit

i =∑−∞<i≤iX πkXit

i. Then, applying π0

to both sides of (6.30),

π0(C(1)) + π0(D)t−1 = π0A. (6.31)

Thus we can solve π0(Ci) and π0D from π0A. For any H ∈ g, the image of (adH)i is

contained in the span of Ui, · · · , Ur−1. Therefore πk (adC)i is identically zero whenever

i > k. Applying πk, (k > 0) to (6.30),

πk(C(1)) + πk(D)t−1 = πk

(k∑i=0

1

i!(adC)i

)A− πk

(k∑i=1

1

(i+ 1)!(adC)i

)C(1). (6.32)

64

The coefficient of ti on the right hand side of (6.32) depends only on finitely many terms

of A and πj(C) for 0 ≤ j < k. So (6.30) can be solved, step by step, for the components

of Ci and D in the basis U0, · · · , Ur−1.

6.6 Formal solution: arbitrary Lie algebra

Lemma 6.6. Let g be an arbitrary finite dimensional Lie algebra. Let A = Aqtq +

Aq−1tq−1 + · · · be an asymptotic series in g. Assume that the leading coefficient Aq is

regular, so that h defined by Equation (6.3) is a Cartan subalgebra. For a constant D ∈ g

and series B = B−1t−1 +B−2t

−2 + · · · and C = Cwtw + Cw−1t

w−1 + · · · write

Y = exp(B) exp(C)tD.

There exists an integer w, series B in g, C in h and a constant D ∈ h such that Y is a

formal solution of

Y (1)(t) = A(t)Y (t). (6.33)

If g′ is another matrix Lie algebra and φ : g → g′ is a Lie algebra isomorphism, then

Y1 = exp(φB) exp(φC)tφD is a formal solution of Y(1)1 = (φA)Y1.

Proof. Let r be the rank of g and let η(X) be the coefficient of λr in the characteristic

polynomial of adX. Since Aq is regular, the subalgebra

h = Y ∈ g : ad(Aq)sY = 0 for some s ≥ 1

is a Cartan subalgebra, nilpotent and with dimension r. For X ∈ h we have det(adX)g/h =

η(X). In particular, det(adAq)g/h = η(Aq) 6= 0.

We look for a formal solution of the form

y ∼ exp(B)h (6.34)

where B is a formal series B ∼ B−1t−1 +B−2t−2 + · · · and h is a formal solution of

h(1) = (Hqtq +Hq−1t

q−1 + · · · )h

where each Hi is in the Cartan subalgebra h. For now the Hi are unknowns. Once we

have chosen them we can find a formal series for h using Lemma 6.5.

The formal derivative of (6.34) is (using Equation (6.12) and Lemma 6.2)

exp(B)

((1− exp(−adB)

adB

)B(1) +H

)h.

Substituting into (6.33) and using (6.22), (6.18) and (6.17), this is equivalent to(1− exp(−adB)

adB

)B(1) +H = exp(−adB)A.

This depends on the structure of g but not on its representation, so if φ : g → g′ is a Lie

algebra isomorphism and Y is a solution of (6.33) then Y1 = exp(φB) exp(φC)tφD is a

solution of Y(1)1 = (φA)Y1. Expanding series,( ∞∑

i=0

(−1)i

(i+ 1)!(adB)i

)B(1) +H =

( ∞∑i=0

1

i!(adB)i

)A. (6.35)

65

Extracting the first two terms of the sum on the right hand side of (6.35), and isolating

H − [B,Aqtq],

H − [B,Aqtq] = −

( ∞∑i=0

(−1)i

(i+ 1)!(adB)i

)B(1) +

( ∞∑i=2

1

i!(adB)i

)A+ [B,A−Aqtq] +A.

(6.36)

Now we match coefficients of powers of t. The coefficient of tq on the left hand side of (6.36)

is Hq, and the coefficient of tq on the right hand side is Aq. So we can match coefficients

by setting Hq = Aq. By Equation (6.3) we know Aq lies in the Cartan subalgebra h. For

a given integer i < q, the coefficient of ti on the left hand side of (6.36) is Hi− [Bi−q, Aq].

The coefficient of ti on the right hand side of (6.36) depends only on Bk−q for k > i.

Therefore we can solve (6.36) for Hi− [Bi−q, Aq], step by step as i decreases starting from

q.

Now, Aq is regular, so that the induced map (adAq)g/h on the quotient g/h is invertible.

It follows that

im adAq + h = g,

i.e. every element of g can be written as [Aq, X] + Y for some X ∈ g and some Y ∈ h.

Thus we can solve (6.36) for Hi and Bi−q, step by step, as i decreases starting from q, and

Hi only depends on finitely many Bj , Aj . Applying Lemma 6.5 to find a formal solution

of h(1) = Hh, we now have a formal solution to (6.33) of the form

exp(B−1t−1 +B−2t

−2 · · · ) exp(Cwtw + Cw−1t

w−1 + · · · )tD,

where all Bi lie in g and all Ci and D lie in a Cartan subalgebra h of g.

66

Chapter 7

Cubics and negative curvature1

Michael Pauley and Lyle Noakes

Abstract

Riemannian cubics are curves that generalise cubic polynomials to arbitrary

Riemannian manifolds, in the same way that geodesics generalise straight lines.

Considering that geodesics can be extended indefinitely in any complete mani-

fold, we ask whether Riemannian cubics can also be extended indefinitely. We

find that there are always exceptions in Riemannian manifolds with strictly

negative sectional curvature.

7.1 Introduction

The problem of interpolating a sequence of points in a Riemannian manifold with a curve

has led to the proposal of Riemannian cubics. They are critical points of the functional∫ T1

T0

〈∇tx(1),∇tx(1)〉dt (7.1)

where ∇ is the Levi-Civita covariant derivative. The Euler-Lagrange equation is

∇3tx

(1) = −R(∇tx(1), x(1))x(1) (7.2)

where R is the Riemannian curvature tensor. Interpolation curves formed from piecewise

Riemannian cubics were independently proposed for applications in computer graphics

in [24], for statistics in [38] (although the paper’s main concern is another interpolation

method), and for robotic control in [57]. The integral (7.1) is one of the simplest functionals

depending on second derivatives – we see Riemannian cubics as a prototype for the study

of higher order variational problems.

Riemannian cubics are higher order geodesics, in the same relation to geodesics as cubic

polynomials are to affine lines. The mathematics of cubics is known to be much richer

than for geodesics, even concerning questions about the basic properties of these curves.

In the present paper, motivated by the large body of work on the long term dynamics of

1Submitted.

67

geodesics (see for example [31, 62]), we answer some simple questions about the long term

dynamics of Riemannian cubics.

We are interested in the solutions of (7.2) for values of t outside of the interval [T0, T1]

specified in the variational problem (7.1). In the rotation group SO(3) equipped with a

bi-invariant metric, geodesics are projective lines, while the long term dynamics of (7.2)

are complicated. (See [52] for the long term dynamics of typical Riemannian cubics in

SO(3), or [51, 56] for a special case about which more is known.)

Any compact semisimple Lie group admits a bi-invariant Riemannian metric, and the

solutions of (7.2) can be extended to the whole real line. This was proven in [52] for SO(3)

but the method there applies in the general case of a compact semisimple Lie group. As

we will see in this paper, in any locally symmetric complete Riemannian manifold whose

sectional curvature is everywhere non-negative, the Riemannian cubics can be extended

to the real line. Surprisingly, the situation is completely different for spaces of negative

sectional curvature.

In an arbitrary complete Riemannian manifold M with a specified point p, one defines

the map expp : TpM →M which takes the vector A ∈ TpM to x(1), where x is the geodesic

with x(0) = p and x(1)(0) = A. If M is locally symmetric with non-negative sectional

curvature, we can do something similar for Riemannian cubics: for a specified p ∈M and

A ∈ TpM let expA : TpM⊕TpM → TM be the map which takes B,C to x(1)(1) where x is

the cubic with x(0) = p, x(1)(0) = A,∇tx(1)(0) = B,∇2tx

(1)(0) = C. On arbitrary M this

map may only be defined on a subset of TpM ⊕ TpM . We will show that if the sectional

curvature is strictly negative, there exist initial conditions for which equation (7.2) does

not have solutions for all t ∈ R. Since cubics stay cubics after the reparametrisation

t → αt (where α ∈ R), there exist initial conditions for which the equations do not have

solutions defined even on the interval [0, 1].

Write K(σ) for the sectional curvature in a tangent plane σ of some point in M . We

say that M has strictly negative sectional curvature if there exists some λ > 0 such that

for any x ∈M and any plane σ in TxM , we have K(σ) ≤ −λ.

The main results of the paper are as follows: In Section 7.3 we prove the main theorem

that in any manifold of strictly negative curvature, initial conditions can be chosen for

Riemannian cubics whose speed diverges to infinity in finite time; thus these cubics cannot

be extended to R. After that we restrict our attention to locally symmetric spaces. In

Section 7.5 we give an example: we find initial conditions for a cubic in the hyperbolic

plane for which 〈x(1), x(1)〉 can be solved exactly; it diverges in finite time. In Section 7.6

we prove that in locally symmetric spaces of non-negative sectional curvature, Riemannian

cubics can be extended to R.

For some applications length may be incorporated into the functional. Riemannian

cubics in tension are critical points of∫ T1

T0

〈∇tx(1),∇tx(1)〉+ τ〈x(1), x(1)〉dt

where τ is a positive real constant. The Euler Lagrange equation is

∇3tx

(1) = −R(∇tx(1), x(1))x(1) + τ∇tx(1). (7.3)

68

Cubics in tension were introduced in [79, 80] (where they are called elastic curves). Their

behaviour in a Lie group with a bi-invariant metric, and in particular in SO(3), is studied

in [58, 59, 80]. The results of Sections 7.3 and 7.6 apply also to Riemannian cubics in

tension.

7.2 Preliminary calculations

In this section we make some calculations that will be needed throughout the paper.

Recall the following identities of the Riemannian curvature tensor R. For X,Y, Z,W

tangent vectors at a point (see for example [71, Chapter 15]):

R(X,Y )Z = −R(Y,X)Z (7.4)

〈R(X,Y )Z,W 〉 = −〈R(X,Y )W,Z〉 (7.5)

〈R(X,Y )Z,W 〉 = 〈R(W,Z)Y,X〉. (7.6)

Let p be a point in a Riemannian manifold M and let X,Y ∈ TpM be linearly independent.

The sectional curvature in the plane σ = spanA,B is

K(σ) =〈R(X,Y )Y,X〉

〈X,X〉〈Y, Y 〉 − 〈X,Y 〉2(7.7)

which, because of the symmetries of R, does not depend upon the choice of X and Y

spanning σ.

Let τ ≥ 0. (We treat (7.2) as a special case of (7.3) with τ = 0.) For any solution

x : (t−, t+)→M of (7.3) and for non-negative integers i, j define

Fij = 〈∇itx(1),∇jtx

(1)〉.

Then we have

Lemma 7.1. Suppose the sectional curvature is bounded above by −λ for some λ > 0.

Then the following equalities and inequalities hold:

F(1)00 = 2F10 (7.8)

F(1)10 = F20 + F11 (7.9)

F(1)20 = τF10 + F21 (7.10)

F(1)11 = 2F21 (7.11)

F(1)21 ≥ λ(F11F00 − F 2

10) + τF11 + F22 (7.12)

F11F00 − F 210 ≥ 0 (7.13)

F22F00 − F 220 ≥ 0 (7.14)

F22F11 − F 221 ≥ 0. (7.15)

Proof. Equations (7.8), (7.9) and (7.11) need only the definition of Fij . Equation (7.10)

follows by differentiating F20, substituting (7.3) and using (7.5) Equations (7.13), (7.14),

(7.15) are all Cauchy-Schwarz inequalities. Equation (7.12) follows by differentiating F21,

substituting (7.3) and (7.7), and using K(σ) ≤ −λ and (7.13).

69

The following standard application of the Mean Value Theorem will be used repeatedly in

the proof of the main theorem. For u, v ∈ Rn write u > v if ui > vi for every i ∈ 1, · · · , n.Also write u ≥ v if ui ≥ vi for every i ∈ 1, · · · , n.

Lemma 7.2. Suppose g, g† : [0, T ) → Rn are continuously differentiable and satisfy the

following condition: whenever g(t) > g†(t), we have also g(1)(t) ≥ g†(1)(t). Suppose too

that g(0) > g†(0). Then g(t) > g†(t) for all t ∈ [0, T ).

Proof. Using the Mean Value Theorem one can show that the sett ∈ [0, T ) : g(t) ≯ g†(t)

has no infimum; thus it is empty.

7.3 Main theorem

Theorem 7.3. Suppose a Riemannian manifold M of dimension at least 2 has strictly

negative sectional curvature. Let T ∈ (0,∞] and x : [0, T ) → M be a solution of (7.3)

satisfying the following constraints on the initial conditions:F00(0) > 0

F10(0) > 0

F20(0) + F11(0) > 0

F21(0) > 0

, (7.16)

(F11F00 − F 210)|t=0 > 0, (7.17)

(F21F00 − F20F10)|t=0 > 0, (7.18)

(F21F10 − F20F11)|t=0 > 0. (7.19)

Then T 6=∞.

Remark. To see that such initial constraints can always be satisfied, choose an arbitrary

x(0) ∈M and an arbitrary orthonormal pair of vectors A,B ∈ Tx(0)M . Let

x(1)(0) = A; ∇tx(1)(0) = A+B; ∇2tx

(1)(0) = A+ 2B.

Then it can be checked that (7.16), (7.17), (7.18), (7.19) hold.

Proof. Let H = F20 + F11. Our strategy is as follows:

Step 1 If (7.16) holds then F00(t) > 0, F10(t) > 0, H(t) > 0, F21(t) > 0 for all t ∈ [0, T ).

Step 2 Let

v = F21F10 − F20F11. (7.20)

If (7.16) and (7.19) then v(t) > 0 for all t ∈ [0, T ).

Step 3 Let

u = F11F00 − F 210. (7.21)

70

If (7.16), (7.17), (7.18), (7.19) then the following system of equations and inequalities

holds:

F(1)00 = 2F10

F(1)10 = H

H(1) = τF10 + 3F21

F(1)21 ≥ λu

u(2) ≥ 2λF00u

with initial constraints

F00(0) > 0

F10(0) > 0

H(0) > 0

F21(0) > 0

u(0) > 0

u(1)(0) > 0

. (7.22)

Step 4 If (7.16), (7.17), (7.18), (7.19), then, using (7.22), there exists T0 > 0 such that

F00 is bounded below by the function 21/λ(T0−t)2 and therefore F00 diverges to infinity

in time T ≤ T0. The theorem follows.

Proof of Step 1. Firstly, from (7.12),

F(1)21 ≥ λ

(F11F00 − F 2

10

)+ τF11 + F22

≥ 0,

by (7.13) and since F11 and F22 are non-negative by definition. From (7.8), (7.9), (7.10),

(7.11) we haveF

(1)00 = 2F10

F(1)10 = H

H(1) = τF10 + 3F21

F(1)21 ≥ 0

with initial constraints

F00(0) > 0

F10(0) > 0

H(0) > 0

F21(0) > 0

. (7.23)

Compare (7.23) with the systemF†(1)00 = 0

F†(1)10 = 0

H†(1) = 0

F†(1)21 = 0

with initial conditions

F †00(0) = 0

F †10(0) = 0

H†(0) = 0

F †21(0) = 0

the solution of which is F †00 = 0, F †10 = 0, H† = 0, F †21 = 0. These systems satisfy the

hypothesis of Lemma 7.2, so it follows that F00 > 0, F10 > 0, H > 0 and F21 > 0 for all

t ∈ [0, T ).

Proof of Step 2. Differentiating (7.20) and using (7.9), (7.10), (7.11), (7.12) and Step 1,

v(1) ≥(λ(F11F00 − F 2

10) + τF11 + F22

)F10 + F21(F20 + F11)

− (τF10 + F21)F11 − 2F20F21

= λ(F11F00 − F 2

10

)F10 + F22F10 − F21F20.

But (7.13) and Step 1 imply that(F11F00 − F 2

10

)F10 is positive, while (7.15) and Step 1

imply

F22F10 − F21F20 ≥F 221

F11F10 − F21F20

=F21

F11v.

71

Thus v(1) > F21F11

v. By Step 1, F21F11

is positive. Now by Lemma 7.2 we can compare v to

the solution v† of v†(1) = 0 with v†(0) = 0. The solution of this is v†(t) = 0 and it follows

that v(t) > 0 for all t ∈ [0, T ).

Proof of Step 3. The first three lines of (7.22) are Equations (7.8), (7.9) and a combi-

nation of (7.10) and (7.11). The fourth line follows from (7.12) because F22 and F11 are

non-negative by definition. Differentiating the defining formula of u,

u(1) = 2F21F00 + 2F11F10 − 2F10(F20 + F11)

= 2(F21F00 − F20F10).

Thus (7.18) implies that u(1)(0) > 0. Differentiating again and using (7.13) and Step 1:

u(2) ≥ 2(λ(F11F00 − F 2

10) + τF11 + F22

)F00 + 4F21F10

− 2(τF10 + F21)F10 − 2F20(F20 + F11)

= 2λF00u+ 2τ(F11F00 − F 210) + 2(F22F00 − F 2

20) + 2v.

By (7.13) and (7.14), we have 2τ(F11F00 − F 210) + 2(F22F00 − F 2

20) ≥ 0, and by Step 2, we

have v > 0. So u(2) ≥ 2λF00u, giving the last line of (7.22).

Proof of step 4. Choose T0 > 0 so large that

21/λ

T 20

< F00(0),21/λ

T 30

< F10(0),63/λ

T 40

< H(0),

84/λ

T 50

< F21(0),420/λ2

T 60

< u(0),2520/λ2

T 70

< u(1)(0).

Compare system (7.22) with the system

F†(1)00 = 2F †10

F†(1)10 = H†

H†(1) = 3F †21

F†(1)21 = λu†

u†(2) = 2λF †00u†.

(7.24)

with initial conditions

F †00(0) = 21/λT 20

F †10(0) = 21/λT 30

H†(0) = 63/λT 40

F †21(0) = 84/λT 50

u†(0) = 420/λ2

T 60

u†(1)(0) = 2520/λ2

T 70

.

This system has as a solution:

F †00 =21/λ

(T0 − t)2, F †10 =

21/λ

(T0 − t)3, H† =

63/λ

(T0 − t)4,

F †21 =84/λ

(T0 − t)5, u† =

420/λ2

(T0 − t)6, u†(1) =

2520/λ2

(T0 − t)7.

Now Lemma 7.2 can be applied to compare the systems (7.22) and (7.24). It follows in

particular that F00 >21/λ

(T0−t)2 and so T < T0.

72

7.4 A simplification in locally symmetric spaces

The manifold M is called locally symmetric when ∇R = 0; this means (see for example

[71, Chapter 2])

∇X (R(Y,Z)W )) = R(∇XY,Z)W +R(Y,∇XZ)W +R(Y,Z)∇XW. (7.25)

As before let Fij = 〈∇ix(1),∇jx(1)〉. The purpose of this section is to replace the equations

of Lemma 7.1 with some simpler equations when M is locally symmetric. We will use these

equations in Section 7.5 to find a Riemannian cubic for which F00 can be solved exactly; and

in Section 7.6 to show that Riemannian cubics can be extended to R in locally symmetric

Riemannian manifolds of non-negative sectional curvature.

Lemma 7.4. When M is a locally symmetric Riemannian manifold, for a Riemannian

cubic in tension, x, there is a constant c ∈ R such that

F22 = −〈R(∇tx(1), x(1))x(1),∇tx(1)〉+ τF11 + c. (7.26)

There is a constant b ∈ R such that the following system of differential equations holds:

F(1)00 = 2F10 (7.27)

F(1)10 =

3

2F11 +

1

2τF00 −

1

2b (7.28)

F(1)11 = 2F21 (7.29)

F(1)21 = −2K(σ)

(F11F00 − F 2

10

)+ 2τF11 + c (7.30)

where K(σ) is the sectional curvature of the plane spanned by x(1) and ∇tx(1).

Proof. Equations (7.27) and (7.29) are just Equations (7.8) and (7.11). Equation (7.28)

follows by combining (7.11), (7.10), (7.9) and (7.8) and integrating. Only (7.26) and (7.30)

require M to be a locally symmetric space. Directly from (7.3) we have

F(1)22 = −2〈R(∇tx(1), x(1))x(1),∇2

tx(1)〉+ 2τF21, (7.31)

while, by (7.25) and (7.4), (7.5), (7.6),

d

dt〈R(∇tx(1), x(1))x(1),∇tx(1)〉 = 2〈R(∇tx(1), x(1))x(1),∇2

tx(1)〉. (7.32)

Combining (7.31) with (7.32) and (7.29), and integrating, we get (7.26). Equation (7.30)

follows since F(1)21 = F31 + F22 = 〈−R(∇tx(1), x(1))x(1) + τ∇tx(1),∇tx(1)〉 + F22, and by

(7.7).

7.5 An example with exact F00

Now let our manifold be the hyperbolic plane H2. It can be defined as follows: Let J be

the 3× 3 diagonal matrix with entries −1, 1, 1. We have a bilinear form 〈·, ·〉 on R3 given

by 〈u, v〉 = uTJv. Let H2 be the set of points x satisfying 〈x, x〉 = −1. The restriction of

〈·, ·〉 to tangent planes of H2 is actually positive definite and we use it as the Riemannian

metric.

73

The hyperbolic plane is a symmetric space, and therefore a locally symmetric space

(Riemannian curvature satisfies (7.25)). In fact the sectional curvature K(σ) is identically

−1. Let τ = 0, so that we are considering plain Riemannian cubics. Choose an arbitrary

point x(0) ∈ H2 and an arbitrary orthonormal basis A,B of Tx(0)H2. Choose the initial

conditions

x(1)(0) =√

10A,

∇tx(1)(0) =√

10A+√

10B,

∇2tx

(1)(0) =√

10A+ 3√

10B.

and suppose that x is a Riemannian cubic with these initial conditions. Then we have

F00(0) = 10, F10(0) = 10, F20(0) = 10,

F11(0) = 20, F21(0) = 40, F22(0) = 100,

and (F11F00 − F 210)|t=0 = 100. Now (7.9) and (7.28) imply that b = 0. Equation (7.30)

with τ set to zero becomes F22 = F11F00−F 210 + c by way of (7.7) with K(σ) = −1. Thus

c = 0. So,

F(1)00 = 2F10, F

(1)10 =

3

2F11, F

(1)11 = 2F21, F

(1)21 = 2(F11F00 − F 2

10).

The solution of this system with the given initial conditions is

F00(t) =10

(1− t)2, F10(t) =

10

(1− t)3,

F11(t) =20

(1− t)4, F21(t) =

40

(1− t)5.

7.6 Locally symmetric spaces with non-negative curvature

The question of extendibility seems more difficult in complete manifolds of non-negative

sectional curvature. At least for complete locally symmetric spaces the question can be

answered. (We return to the general case of a cubic with or without tension i.e. τ ≥ 0, b

and c are arbitrary.)

Theorem 7.5. Let M be a complete locally symmetric Riemannian manifold such that

K(σ) ≥ 0 for any tangent plane σ at any point of M . Then any solution of (7.3) can be

extended to R.

Proof. Suppose that x, x(1),∇tx(1),∇2tx

(1) are defined at some time t0. First we find

functions that bound Fii above (i = 0, 1, 2) for t > t0. By (7.13) and the non-negative

curvature, the term −2K(σ)(F11F00−F10) in (7.30) is non-positive. Compare the system

of equations (7.27), (7.28), (7.29), (7.30) with a new system formed by deleting this non-

positive term: F†(1)00 = 2F †10

F†(1)10 = 3

2F†11 + 1

2τF†00 − 1

2 b

F†(1)11 = 2F †21

F†(1)21 = 2τF †11 + c

(7.33)

74

with initial constraints F †00(t0) > F00(t0)

F †10(t0) > F10(t0)

F †11(t0) > F11(t0)

F †21(t0) > F21(t0)

.

By Lemma 7.2, the functions F00, F10, F11, F21 are bounded above by the functions F †00,

F †10, F†11, F

†21 respectively. But (7.33) is a linear system of ODEs, so the solutions exist

for all time. Equation (7.26) implies that F22 ≤ τF11 + c < τF †11 + c.

Let

S = T : the cubic x can be extended to [t0, T ] .

If S is bounded above, let T = supS. Then since F †00, F†11 exist for all time, and are

continuous, we can define

µ = maxF †00(t), F

†11(t), τF

†11(t) + c : t ∈ [t0, T ]

.

Then for any T ∈ [t0, T ), if x is defined on [t0, T ], then the functions F00, F11, F22 are

bounded above by µ for all t ∈ [t0, T ]. Therefore ‖x(1)‖ ≤ √µ so x(T ) is contained in a

closed ball in M , of radius (T−t0)√µ centred at x(t0). Also x(1)(T ), ∇tx(1)(T ), ∇2

tx(1)(T )

are contained in a closed ball in Tx(T )M , of radiusõ centred at 0.

Picard’s theorem tells us that there is δ > 0 and a unique solution x : (T−δ, T+δ)→M

of (7.3) with the initial conditions x(T ), x(1)(T ), ∇tx(1)(T ), ∇2tx

(1)(T ). In fact, since for

t0 ≤ T < T , the initial conditions are restricted to a compact set, we can choose δ

independently of T . By choosing T = T − 12δ we see that x can be extended to [0, T + 1

2δ).

This contradicts the claim that T = supS.

This shows that x can be extended to the interval [t0,∞). Applying the same argument

to x(−t), which is also a cubic in tension, shows that x can be extended to (−∞,∞).

75

76

Chapter 8

Geodesic curvature and asymptotic

geodesics

8.1 Introduction

Let M be a complete smooth Riemannian manifold, t0 ∈ R and let y : [t0,∞) → M be

a smooth, arc-length parametrised curve, i.e. ‖y(1)(t)‖ = 1 for all t. Write ∇tX for the

time-derivative of a vector field X along y, given by the Levi-Civita covariant derivative.

The (absolute) geodesic curvature of y is the function

k(t) = ‖∇ty(1)(t)‖.

How small, on average, must the geodesic curvature be as t → ∞ in order to ensure the

existence of some geodesic γ(t) such that d(y(t), γ) −−−→t→∞

0? By “small on average” we

mean the existence of a function f(t) such that∫∞t0f(t)k(t)dt is finite; the larger f is

asymptotically, the “smaller on average” k is.

Enomoto studied this question in [23], and found that when M is Euclidean space,∫ ∞t0

r(t)k(t)dt converges (8.1)

implies the existence of asymptotic geodesics, where r(t) = d(origin, y(t)). Enomoto found

that, surprisingly, if M is hyperbolic space, the (typically weaker) condition∫ ∞t0

k(t)dt converges (8.2)

is sufficient. The sufficiency of (8.2) appears to be related to a fact about geodesics in the

hyperbolic plane: given any geodesic γ1 and any point p, there exists a unique geodesic γ2

through p such that d(γ1(t), γ2) converges to zero as t→∞. This fact is actually known

to be true in any space of strictly negative curvature, i.e. any Riemannian manifold in

which the sectional curvature is bounded above by some negative constant [9, Lemmas

9.1, 10.7]. Thus we may expect that, in any space of strictly negative curvature, (8.2)

implies the existence of an asymptotic geodesic. The purpose of this chapter is to show

that this is true. We also show that (8.2) implies the existence of an asymptotic geodesic

in the n-sphere. Lastly we apply the result to give asymptotic results on solutions of

certain ordinary differential equations which arise in an interpolation problem.

We find ideas from [9, Sections 9-10] helpful.

77

γ(t)

y(t)

q(u,t)

q(1,t)

q(0,t)γ(t)

y(t)

α(0,t)

α(u,t)

α(1,t)

Figure 8.1: Definitions of q and α defined in Lemma 8.1 and used throughout the

chapter.

8.2 Some lemmas in simply connected manifolds of non-positive curva-

ture

In this section let M be a simply connected Riemannian manifold such that K(X,Y ) ≤ 0.

Let I be a subinterval of R. Let y : I → M be a smooth curve and let γ : I → M be a

geodesic.

Lemma 8.1. (See Figure 8.1.) Define q : [0, 1]× I →M by letting q(·, t) be the geodesic

between q(0, t) = γ(t) and q(1, t) = y(t). Then q is uniquely defined and smooth.

Define α : [0, 1]× I →M by letting α(·, t) be the shortest geodesic between α(0, t) ∈ γand α(1, t) = y(t). Then α is also uniquely defined and smooth.

Proof. (α.) The exponential mapping exp : TM →M is smooth. Therefore the fact that

α is uniquely defined and smooth follows from the fact that the exponential map gives a

diffeomorphism from the normal bundle of γ onto M . (See [9, Lemma 3.1].)

(q.) By the Cartan-Hadamard Theorem, q is uniquely defined. Fix t, and let A(t) ∈Tγ(t)M be such that expγ(t)(A) = y(t). The Levi-Civita connection allows us to identify

the derivative (d expγ(t))A with the map which, for any Jacobi field J along the geodesic

v 7→ expγ(t)(vA), maps J(0),∇tJ(0) to J(1). (If z : R → TM is a smooth curve in TM

then we can treat z as a curve x in M coupled with a tangent vector field A along x. At

a given time t we can identify the tangent vector z(1)(t) with the pair (x(1)(t), (∇tA)(t))

of tangent vectors of M at x(t).) From the Cartan-Hadamard Theorem, expγ(t) is a

diffeomorphism Tγ(t)M →M . So the restriction of (d expγ(t))A to the Jacobi fields J with

J(0) = 0 is an invertible linear map. Therefore the restriction of (d expγ(t))A to the Jacobi

fields J with J(0) = γ(1)(t) is onto. It follows from the inverse function theorem that

A(t) is smooth. Since exp, γ, A are smooth, and q(u, t) = expγ(t)(uA(t)), we see that q is

smooth.

Lemma 8.2. Let I be a subinterval of R. Let y : I → M be a smooth curve, and let

γ : I → M be a geodesic. Let G(t) = d(y(t), γ(t)). Then G is continuous on I, and the

times t ∈ I fall into three cases:

1. t such that G(t) > 0, and G is smooth in a neighbourhood of t;

2. t such that G(t) = 0 and G is twice differentiable at t with G(2)(t) ≥ 0;

3. t such that G(t) = 0 and G(1) has a positive jump discontinuity at t, namely

limv→t+ G(1)(t)− limv→t− G

(1)(t) > 0.

78

Let L(t) = d(y(t), γ). Then L is also continuous and the times t ∈ I fall into the three

cases described above, with L in place of G.

Proof. With q and α defined as in Lemma 8.1, we have for any t, u that

G(t) =√〈qu(u, t), qu(u, t)〉

and L(t) =√〈αu(u, t), αu(u, t)〉. Since q and α are smooth, and

√· is continuous, and

smooth except at zero, we see that G and L are smooth at any point where they are

nonzero. This takes care of the first case. Suppose t is such that G(t) = 0, i.e. y(t) =

γ(t). Since G is non-negative, limv→t+ G(1)(v) is non-negative and limv→t− G

(1)(v) is

nonpositive. Therefore limv→t+ G(1)(v) − limv→t− G

(1)(v) ≥ 0. We now split into two

cases:

• y and γ meet transversally at t. Then t is an isolated zero of G, and limv→t+ G(1)(v)−

limv→t− G(1)(v) > 0.

• y and γ meet tangentially at t. Then G(1) is continuous at t. Write H(v) = G(v)2 =

〈qu(u, v), qu(u, v)〉 for arbitrary u. Since H is non-negative, and by Lemma 8.1,

smooth, we have H(t) = 0, H(1)(t) = 0. Since G is non-negative and G(1) is continu-

ous at t, we must have G(1)(t) = 0 so H(2)(t) = 0. But then, since H is smooth and

non-negative, H(3)(t) is also zero, and H(4)(t) ≥ 0. By Taylor’s Theorem, H(v) =H(4)(t)

4! (v − t)4 + o(v − t)4 as v → t. Therefore G(v) = +

√H(4)(t)

4! (v − t)2 + o(v − t)2

as v → t, that is, G is twice differentiable at t and G(2)(t) ≥ 0.

A similar argument applies to L.

Remark. It is necessary to say “twice differentiable” and not “smooth” in Case 2 of Lemma

8.2. Consider, for example, when M is the Euclidean plane E2 and y : t 7→ (t, t3), γ : t 7→(t, 0). Then G(t) = L(t) = |t3|, which is twice differentiable at t = 0 but not smooth, and

G(1)(t) = L(1)(t) = 3t|t| does not have a jump discontinuity at t = 0.

Lemma 8.3. Let y : I → M be a smooth curve, parametrised by arc-length, and let

k(t) = ‖∇tyt‖ be the geodesic curvature. Let k2(t) be such that k(2)2 (t) = k(t). Then

t 7→ d(y(t), γ(t)) + k2(t)

is a convex function of t.

Proof. Let G(t) = d(y(t), γ(t)) so that G(t)2 = 〈qu, qu〉 as in Lemma 8.2. As in [9, Section

10], differentiating in t and using ∇uqu ≡ 0,

GG(1) = 〈∇tqu, qu〉 =∂

∂u〈qt, qu〉,

so that (differentiating in t again)

GG(2) =∂

∂u〈∇tqt, qu〉+ 〈∇tqu,∇tqu〉+ 〈qt,∇u∇tqu〉 − (G(1))2.

79

Computing 〈∇tqu,∇tqu〉 − (G(1))2 = 〈∇tqu,∇tqu〉 − 〈∇tqu,qu〉2〈qu,qu〉 = ‖∇tqu∧qu‖2

‖qu‖2 ≥ 0 and

〈qt,∇u∇tqu〉 = −K(qt, qu)‖qt ∧ qu‖2 ≥ 0, we have

GG(2) ≥ ∂

∂u〈∇tqt, qu〉.

The left hand side does not depend on u. Integrating both sides from u = 0 to u = 1,

GG(2) = (1− 0)GG(2) ≥∫ 1

0

∂

∂u〈∇tqt, qu〉du

= [〈∇tqt, qu〉]u=1u=0.

Now, 〈∇tqt, qu〉|u=0 = 0 since q(0, t) = γ(t) is a geodesic. By the Cauchy-Schwarz inequal-

ity, |〈∇tqt, qu〉|u=1| ≤ ‖∇ty(1)‖‖qu‖ = k(t)G(t). So

G(G(2) + k

)≥ 0. (8.3)

By Lemma 8.2 the times t ∈ I fall into three cases:

• G(t) is nonzero. Then from (8.3) we have G(2)(t) + k(t) ≥ 0.

• G(t) = 0 and G is twice differentiable at t with G(2)(t) ≥ 0. Then, since k(t) ≥ 0 we

have G(2)(t) + k(t) ≥ 0.

• t is an isolated positive jump discontinuity of G(1). Then, since k(1)2 is continuous,

G(1) + k(1)2 has a positive jump discontinuity at t.

Combining all three cases, it follows that G(1)+k(1)2 is nondecreasing; therefore G(t)+k2(t)

is convex.

Remark. A similar argument would show that d(y(t), γ) + k2(t) is also a convex function

of t. We do not need this; in strictly negative curvature we will prove a stronger property

of d(y(t), γ) + k2(t), as part of the proof of the main theorem.

Lemma 8.4. Let y : [t0,∞) → M be a smooth curve, parametrised by arc-length, such

that the geodesic curvature k(t) = ‖∇tyt‖ satisfies (8.2). Then, through any p ∈ M there

is a geodesic γ such that d(y(t), γ(t)) = o(t). It follows that d(y(t), γ) = o(t).

Proof. Define

k1(t) =

∫ t

∞k(τ)dτ which exists by the hypothesis; (8.4)

k2(t) =

∫ t

t0

k1(τ)dτ. (8.5)

Let ti be a sequence of times increasing to∞ as i→∞. Let γi be the geodesic (unique by

Cartan-Hadamard) with γi(t0) = p and γi(ti) = y(ti). From [23, Lemma 1.1] it is known

that, as t→∞,

d(y(t), p) = t− o(t). (8.6)

It follows that the sequence of tangent vectors Ai = γ(1)i (t0) ∈ TpM satisfies ‖Ai‖ −−−→

i→∞1.

Therefore (Bolzano-Weierstrass) some subsequence of the Ai converges to some A ∈TpM , and in fact ‖A‖ = 1. Let γ(t) = expp((t− t0)A).

80

By Lemma 8.3, for each of the geodesics γi, the function d(y(t), γi(t))+k2(t) is a convex

function of t. Therefore, for t0 ≤ t ≤ ti,

d(y(t), γi(t)) + k2(t) ≤t− t0ti − t0

(d(y(ti), γi(ti)) + k2(ti)) +ti − tti − t0

(d(y(t0), γi(t0)) + k2(t0))

=t− t0ti − t0

(0 + k2(ti)) +ti − tti − t0

(d(y(t0), p) + 0)

≤ t− t0ti − t0

k2(ti) + d(y(t0), p).

But since k is a positive function, and by the definition of k1, we have k1 negative, so k2

is everywhere decreasing. Now k2(t0) = 0 so k2(ti) ≤ 0, and

d(y(t), γi(t)) ≤ d(y(t0), p)− k2(t).

Since exp is continuous, and by the definition of γ, we have

d(y(t), γ(t)) ≤ d(y(t0), p)− k2(t).

Now d(y(t0), p) does not depend on t, and k2(t) = o(t) since its derivative k1(t) converges to

zero as t→∞. Therefore d(y(t), γ(t)) = o(t) as t→∞. Since 0 ≤ d(y(t), γ) ≤ d(y(t), γ(t))

we also have d(y(t), γ) = o(t).

8.3 The theorem and its proof

Theorem 8.5. Let M be a Riemannian manifold with strictly negative sectional curvature.

Let y : [t0,∞)→M be a smooth curve parametrised by arc-length, with geodesic curvature

k(t). Suppose∫∞t0k(t)dt is finite. Let p ∈ M . Then there is a geodesic γ through p, such

that d(y(t), γ) −−−→t→∞

0.

Proof. Since a Riemannian covering mapping does not increase distance, we can assume

without loss of generality that M is simply connected (as in [9, Lemma 10.7]). Let λ > 0

such that K(X,Y ) ≤ −λ for all tangent vectors X,Y at all points of M . Define the

functions k1, k2 as in Equations (8.4), (8.5). Let γ be a geodesic through p such that

d(y(t), γ) = o(t) as t→∞, as in Lemma 8.4. We prove the theorem in the following steps:

Step 1 Let L(t) := d(y(t), γ). With α defined as in Lemma 8.1, let

F (u, t) := ‖αu ∧ αt‖2. (8.7)

Then

Fuu(u, t) ≥ 2λL(t)2F (u, t), (8.8)

so that Fu can be bounded above using the values of F |u=1 and Fu|u=1:

Fu ≤ −(F |u=1)√

2λL sinh(√

2λL(1− u))

+ (Fu|u=1) cosh(√

2λL(1− u)). (8.9)

Step 2 Computing the values of F |u=1, Fu|u=1 and Fu|u=0, we can conclude using Equa-

tion (8.9) that

L(2) + k ≥ (1− (L(1))2)√λ/2 tanh(

√2λL), (8.10)

wherever the left hand side exists. When the left hand side does not exist, L(1) + k1

has a positive jump discontinuity.

81

Step 3 The derivative Lt satisfies

L(1) −−−→t→∞

0. (8.11)

Combining (8.10) and (8.11) we can show that

M(t) :=

∫ t

∞

√λ/8 tanh(

√2λL(τ))dτ, (8.12)

exists, and

M,M (2) −−−→t→∞

0,

so that M (1) −−−→t→∞

0. It follows that L −−−→t→∞

0.

Proof of Step 1. Since, for fixed t, α(·, t) is a geodesic, for any u, t we have L(t)2 =

〈αu, αu〉. Differentiating (8.7) in u, and using ∇uαu ≡ 0,

Fu = 2〈αu ∧∇uαt, αu ∧ αt〉 (8.13)

Differentiating again, and using ∇uαu ≡ 0,

Fuu = 2〈αu ∧∇2uαt, αu ∧ αt〉+ 2〈αu ∧∇uαt, αu ∧∇uαt〉

= 2〈αu ∧R(αu, αt)αu, αu ∧ αt〉+ 2〈αu ∧∇uαt, αu ∧∇uαt〉

= 2 (〈αu, αu〉〈R(αu, αt)αu, αt〉 − 〈αu, αt〉〈R(αu, αt)αu, αu〉) + 2〈αu ∧∇uαt, αu ∧∇uαt〉.

Since 〈R(·, ·)·, ·〉 is skew-symmetric in the last two parameters, the second term is zero,

and, using 〈R(X,Y )Y,X〉 = K(X,Y )‖X ∧ Y ‖2,

Fuu ≥ −2K(αu, αt)〈αu, αu〉‖αu ∧ αt‖2

≥ 2λL2F. (8.14)

For fixed t, let F † : [0, 1]→ R be the solution of

F †uu = 2λL2F †, (8.15)

with initial conditions F †|u=1 = F |u=1 and F †u|u=1 = Fu|u=1. It can be checked that F †u

is the right hand side of (8.9). Comparing Equations (8.14) and (8.15) using Lemma 7.2,

we see that Fu < F †u for 0 ≤ u ≤ 1.

Proof of Step 2. Since γ is a geodesic, and α(0, ·) is a reparametrisation of γ, we have

αt|u=0 and ∇tαt|u=0 linearly dependent. Since α(·, t) meets α(0, ·) at a right angle, we

have

〈αu, αt〉|u=0 = 0; (8.16)

〈αu,∇tαt〉|u=0 = 0. (8.17)

Therefore, by (8.13),

Fu|u=0 = 2 (〈αu, αu〉〈∇uαt, αt〉 − 〈αu, αt〉〈∇uαt, αu〉)|u=0

= 2

(〈αu, αu〉

(∂

∂t〈αu, αt〉 − 〈αu,∇tαt〉

)+ 0

)∣∣∣∣u=0

,

82

which, by (8.16), (8.17), is zero.

To compute F and Fu at u = 1, note first that

〈∇tαu, αu〉 =∂

∂u〈αt, αu〉 − 〈αt,∇uαu〉 =

∂

∂u〈αt, αu〉,

so that differentiating L2 = 〈αu, αu〉 in t we get

LL(1) =∂

∂u〈αt, αu〉. (8.18)

The left hand side of (8.18) is independent of u, so, integrating from u = 0 to u = 1, we

have, using (8.16),

LL(1) = (1− 0)LL(1) = [〈αt, αu〉]u=1u=0 = 〈αt, αu〉|u=1. (8.19)

Now,

F |u=1 =(〈αu, αu〉〈αt, αt〉 − 〈αu, αt〉2

)|u=1

=(L2 · 1− (LL(1))2

)= L2(1− (L(1))2);

Fu|u=1 = 2 (〈αu, αu〉〈∇uαt, αt〉 − 〈αu, αt〉〈∇uαt, αu〉) |u=1

= 2

(L2

(∂

∂t〈αu, αt〉 − 〈αu,∇tαt〉

)− LL(1) · 1

2

(∂

∂t〈αu, αu〉

))∣∣∣∣u=1

.

To this we apply (8.19),

F |u=1 = 2

(L2

(∂

∂t(LL(1))− 〈αu,∇tαt〉

)− LL(1) · 1

2

∂

∂t(L2)

)∣∣∣∣u=1

= 2L2(LL(2) − 〈αu,∇tαt〉)|u=1.

Since |〈αu,∇tαt〉| ≤ ‖αu‖‖∇tαt‖ = Lk we have

Fu|u=1 ≤ 2L3(L(2) + k).

Substituting u = 0 in (8.9), we have

0 = Fu|u=0 ≤ −(F |u=1)√

2λL sinh(√

2λL)

+ (Fu|u=1) cosh(√

2λL)

≤ L3(−√

2λ(1− (L(1))2) sinh(√

2λL)

+ 2(L(2) + k) cosh(√

2λL))

. (8.20)

By Lemma 8.2 we split times t ∈ [t0,∞) into three cases:

• L(t) 6= 0. Then (8.10) follows by rearranging (8.20);

• L(t) = 0 and L is twice differentiable at t with L(2)(t) ≥ 0. Then the left hand side

of (8.10) is non-negative while the right hand side is zero;

• L(t) = 0 and L(1) has a positive jump discontinuity. Then since k1 is continuous,

L(1)(t) + k1(t) has a positive jump discontinuity.

In all three cases we have proved Step 2.

Proof of Step 3. We first prove (8.11). Since L = o(t) as t → ∞ by the choice of γ,

and since k2 = o(t), we have L+ k2 = o(t). But its derivative, L(1) + k1, is increasing, and

therefore must converge to zero. Since k1 converges to zero, L(1) does also.

83

Now, choose T > t0 such that L(1) < 12 for all t > T . Then from (8.10) we have

L(2)(t) + k(t) ≥√λ/8 tanh(

√2λL(t)), (8.21)

whenever the left hand side exists. Whenever the left hand side does not exist, L(1) + k1

has a positive jump discontinuity. Let

M(t) =

∫ t

T

√λ/8 tanh(

√2λL(τ))dτ. (8.22)

We see by integrating (8.21) that

(L(1)(t) + k1(t))− (L(1)(T ) + k1(T )) ≥ M(t).

Since the left hand side is bounded above, the right hand side is also. But since the

integrand of (8.22) is non-negative, M is a non-decreasing function of t. So M converges

as t → ∞. Therefore we can replace the lower limit of (8.22) with ∞, showing that M

defined by Equation (8.12) exists, and M −−−→t→∞

0. Lastly, compute

M (2)(t) =λ/2

cosh(√

2λL(t))2L(1)(t).

Since L(1)(t) −−−→t→∞

0 and 1cosh2

is a bounded function, we have M (2)(t) −−−→t→∞

0. It is known

[47, pp54-55] that if M → 0 and M (2) is bounded as t→∞ then M (1) → 0. Therefore,

tanh(√

2λL(t)) −−→t→0

0,

which implies L(t)→ 0 as t→∞.

8.4 Condition (8.2) is sufficient in the n-sphere

We consider in this section the n-sphere Sn with the metric induced from n+1 dimensional

Euclidean space.

Proposition 8.6. Let y : [t0,∞)→ Sn ⊆ Rn+1 be a smooth arc-length parametrised curve

whose geodesic curvature k satisfies (8.2). Then there exists a geodesic γ in Sn such that

d(y(t), γ)→ 0 as t→∞.

Proof. By treating any unit tangent vector A at any point p of Sn as the pair (p,A) we

see that the unit tangent sphere bundle T1Sn of Sn is a compact submanifold of (Rn+1)2.

Let φ : (Rn+1)2 → Λ2(Rn+1) be the linear map which takes the pair (v, w) to the bivector

v ∧ w.

Let z : [t0,∞) → Λ2(Rn+1) be the curve defined by z(t) = φ(y(t), y(1)(t)) = y(t) ∧y(1)(t). Knowing that the Levi-Civita connection on the sphere assigns to a vector field

X(t) (tangent to Sn) along y the vector field ∇tX(t) = X(1)(t) − 〈X(1)(t), y(t)〉y(t) =

X(1)(t)+〈X(t), y(1)(t)〉y(t), we have y(2)(t) = ∇ty(1)(t)−〈y(1)(t), y(1)(t)〉y(t) = ∇ty(1)(t)−y(t). Therefore z(1)(t) = y(t) ∧ (∇ty(1)(t) − y(t)) = y(t) ∧ ∇ty(1)(t). Since y(t),∇ty(1)(t)are orthogonal vectors in Euclidean space, ‖z(1)(t)‖ = ‖y(t)‖‖∇ty(1)(t)‖ = k(t). Equation

(8.2) now tells us that∫∞t0‖z(1)(t)‖dt is finite. We will now see that this implies z(t)

84

converges to a point as t → ∞.1 Choose a sequence ti∞i=1 increasing to infinity. For

0 < i < j, the (Euclidean) distance d(z(ti), z(tj)) is bounded above by the length of the

curve between these times,∫ tjti‖z(1)(t)‖dt. Since the integrand ‖z(1)(t)‖ is non-negative,

we have

d(z(ti), z(tj)) ≤∫ ∞ti

‖z(1)(t)‖dt −−−→i→∞

0.

It follows that z(ti)∞i=1 is a Cauchy sequence, so it converges to some z∞ ∈ Λ2(Rn+1).

Now for any t > t0,

d(z(t), z∞) ≤∫ ∞t‖z(1)(t)‖dt

which converges to zero as t→∞. It follows that z(t) −−−→t→∞

z∞. Since T1Sn is compact,

the image φ(T1Sn) is also compact, so z∞ is the image of some pair (p,A) with p ∈ Sn, A ∈

TpSn. Now, p,A ∈ Rn+1 span some 2-dimensional subspace σ of Rn+1 whose intersection

with Sn is the geodesic γ through p with direction A. The distance (in Sn) from any point

x to γ is a continuous function of x and p∧A. Therefore, since y(t)∧ y(1)(t) converges to

p ∧A as t→∞, the distance from y(t) to γ converges to zero.

8.5 Application to JK-cubics

JK-cubics satisfy the ordinary differential equation

∇3t y

(1)(t) = 0. (8.23)

We could generalise to polynomials of other degrees by replacing (8.23) with

∇m−1t y(1)(t) = 0 (m ≥ 2). (8.24)

Thanks to Theorem 8.5 we can say that in manifolds of strictly negative sectional curva-

ture, solutions of (8.24) have asymptotic geodesics.

Corollary 8.7. Let M be a complete Riemannian manifold with strictly negative sectional

curvature and let y : R→M be a solution of (8.24). Then there is a geodesic γ such that

d(y(t), γ) −−−→t→∞

0.

Proof. Instead of finding a formula for the curvature of y we consider a development of

y from M onto Euclidean space. Suppose M has dimension n. Choose an orthonormal

basis Bini=1 for Ty(t0)M and parallel translate each basis element along y, to give a basis

Bi(t)ni=1 for Ty(t)M for every t. Then y(1)(t) can be written in terms of this basis:

y(1)(t) =∑f iBi. Let Eini=1 be an orthonormal basis of n-dimensional Euclidean space.

A development x(t) of y(t) in Euclidean space is a curve x : R → Rn such that x(1)(t) =∑f iEi. The curvature of the development x is the same as the geodesic curvature of the

original curve y. When y satisfies the differential equation (8.24), then the development

1This argument was provided to me by Prof Lyle Noakes in personal communication. It is an elaboration

of an argument in the proof of [52, Theorem 3].

85

x satisfies x(m)(t) = 0. Unless y is the constant map, x is a polynomial of degree at least

1, so the unit tangent vector x(1)(t)

‖x(1)(t)‖ converges as t→∞. Of course, if s is an arc-length

parameter for x thendxds

(s)

‖dxds

(s)‖ still converges as s → ∞. So, choosing so ∈ R we have

that∫∞s0k(s)ds converges, because for arc-length parametrised curves in Euclidean space,

curvature is the rate of change of the unit tangent vector. Since M has strictly negative

curvature, Theorem 8.5 implies that y has an asymptotic geodesic.

86

Chapter 9

Conclusions

In this thesis we studied two generalisations of cubics to affine connection spaces. The

first generalisation is the Riemannian cubic, a solution of the differential equation

∇3tx

(1) +R(∇tx(1), x(1))x(1) = 0. (9.1)

In a Lie group, the differential equation (9.1) is related by Lie reduction to a differential

equation in the corresponding Lie algebra,

V (2) = [V (1), V ] + C (9.2)

where C is a constant in the Lie algebra. A solution of (9.2) is called a Lie quadratic.

When C = 0 the Lie quadratic is called null. The second generalisation is the JK-cubic,

a solution of

∇3tx

(1) = 0. (9.3)

The affine connection spaces of interest to us are Riemannian manifolds where the con-

nection ∇ is the Levi-Civita connection, and Lie groups where the connection of interest

is the (0)-connection.

In Chapter 3 we looked at the method of Lax constraints given in [54]. In semisimple

Lie groups, if Lax constraints are known, [54] gave a way to solve a curve in a Lie group

in quadrature from its Lie reduction. This applies to Riemannian cubics as well as Rie-

mannian cubics in tension and higher order geodesics. In Chapter 3 we showed how the

method of [54] can be modified to apply to arbitrary Lie groups. (Theorem 3.2).

In Chapter 4 we studied a special case of JK-cubics, a solution of

∇2tx

(1) = 0.

We call such a curve a JK-quadratic. In a Lie group with the (0)-connection, a generic

JK-quadratic can be solved in terms of a pair of null Lie quadratics. (Proposition 4.1,

4.2). In the Lie groups SO(3) and SO(1, 2), we solve for generic JK-quadratics and null

Lie quadratics in terms of a quantum harmonic oscillator (Theorem 4.5). Solutions are

related to JK-quadratics and Euler spirals in the sphere and the hyperbolic plane (Sections

4.8,4.7).

In Chapter 5 we studied JK-cubics in n × n matrix groups with the (-) and (0)-

connections. Equation (9.3) with the (-) connection immediately becomes a linear system

87

of ordinary differential equations with polynomial coefficients (Section 5.2.1) and this

allows us to give asymptotic expansions in a generic case in GL(n) (Section 5.3). For the

(0)-connection, Equation (9.3) can be solved in terms of a pair of these systems (Theorem

5.6) which allows us to give an asymptotic description for JK-cubics in GL(n) with the (0)-

connection (Corollary 5.7). This asymptotic description can also be applied to JK-cubics

in the sphere (Section 5.5).

In Chapter 6, motivated by the generic result for GL(n) in Chapter 5, and hoping to

generalise this to other Lie groups, we looked at the way we write asymptotic expansions

for solutions of linear ordinary differential equations. The usual way [84, Chapter IV] gives

an asymptotic solution of

y(1)(t) = (Aqtq +Aq−1t

q−1 + · · · )y(t)

under the generic condition that the leading coefficient Aq has distinct eigenvalues. Asymp-

totic solutions are possible but much more difficult when Aq does not have distinct eigen-

values. When the coefficients Ai lie in some Lie algebra g, it is possible that the condition

of distinct eigenvalues is not generic. The result of Chapter 6 was a way of writing asymp-

totic solutions, which allows us to replace this condition with the condition that Aq is

a regular element of g (Theorem 6.1). This condition is generic in any Lie algebra. We

used this to give an asymptotic description of generic null Riemannian cubics and generic

JK-cubics in a Lie group with the (0)-connection (Sections 6.2 and 6.3).

In Chapters 7 and 8 we turned our attention to manifolds of strictly negative sectional

curvature. A fundamental result on geodesics is that in any complete Riemannian mani-

fold, any geodesic can be extended to the whole real line. We asked in Chapter 7 whether

Riemannian cubics can also be extended indefinitely. It turns out that this is not the case.

In any manifold of strictly negative sectional curvature, there exists an open family of

Riemannian cubics for which the speed diverges to infinity in finite time, thus, they can-

not be extended to the whole real line (Theorem 7.3). In locally symmetric manifolds of

non-negative sectional curvature this phenomenon does not occur – all Riemannian cubics

are extendible (Theorem 7.5).

In Chapter 8, in order to give an asymptotic description of JK-cubics in manifolds of

strictly negative sectional curvature, we generalised a result of Enomoto [23] which was for

curves in the hyperbolic plane. If the geodesic curvature k(t) of an arc-length parametrised

curve y satisfies∫∞t0k(t)dt, then there is a geodesic γ such that d(y(t), γ)→ 0 as t→∞.

We showed that this is true in arbitrary manifolds of strictly negative sectional curvature

(Theorem 8.5). We also proved that this is true in a sphere (Proposition 8.6). We used

Theorem 8.5 to show that JK-cubics (in fact, solutions of ∇k−1t y(1) = 0 for arbitrary k ≥ 2)

in Riemannian manifolds are asymptotically approximated by reparametrised geodesics

(Corollary 8.7).

88

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Index

m-geodesics, 26

ad-invariance, 16

adjoint representation

of a Lie algebra, 12

of a Lie group, 14

affine connection space, 6

affine Riemannian cubic, 24

affinity, 15

asymptotic expansion, 55

bi-invariant semi-Riemannian metric, 16

Cartan subalgebra, 12

Cartan-Hadamard Theorem, 11

Cartan-Killing form, 12

connection

affine, 5

bi-invariant, 15

canonical, mean or (0)-, 14

left invariant, 14

left or (-)-, 15

Levi-Civita, 7

metric, 7

right or (+)-, 15

covariant acceleration, 7

covariant derivative, 6

along a curve, 6

cubic spline, 20

curvature tensor, 6

derivations, 12

development, 27, 85

Engel’s theorem, 60

Euler elastic curve, 26

Euler spiral, 40

exponential mapping, 7, 13

geodesic, 1, 7

geodesic curvature, 4, 22, 28, 77

homomorphism of Lie algebras, 12

hyperbolic plane, 4, 10

ideal, 12

isomorphism of Lie algebras, 12

Jacobi field, 11

JK-cubic, 2, 21, 34

JK-quadratic, 3, 27, 33, 34

Killing form, 12

Lax constraint, 25, 29

Lax equation, 25

left invariant semi-Riemannian metric, 15

Lie algebra, 11

abelian, 12

nilpotent, 12

semisimple, 12

simple, 12

Lie bracket, 11

Lie group, 13

Lie quadratic, 24

affine, 24, 57

dual, 24

in tension, 26

null, 24, 34, 57

Lie reduction, 15, 29

matrix group, 13

minimal geodesic, 9

natural cubic spline, 20

parallel, 7

parallel translation, 7

95

prototype, 29

pseudo-Riemannian metric, 7

quantum harmonic oscillator, 3, 38

rank

of a Lie algebra, 12, 29

regular, 12, 29, 56

Riemannian cubic, 2, 21, 34, 57, 67

affine, 57

null, 57

Riemannian cubic in tension, 26

Riemannian curvature tensor, 6

Riemannian manifold, 8

Riemannian metric, 8

sectional curvature, 9, 69

strictly negative, 68

semi-Riemannian manifold, 7

semi-Riemannian metric, 7

special unitary group, 17

symmetric tensor, 6

tensor, 5

tensor field, 5

torsion, 6

torsion-free, 6

vector field

left invariant, 13

96