Superconnections and a Finslerian Gauss-Bonnet-Chern...

Superconnections and a FinslerianGauss-Bonnet-Chern formula

Huitao Feng (Joint with Ming Li)

Chern Institute of Mathematics, Tianjin, China

2016-08-10, KrakowGlances@Manifolds II

Huitao Feng, Joint with Ming Li (CIM ) Superconnections and a Finslerian GBC-formula 10-08-2016, Krakow 1 / 44

Outline

In this talk, we will establish a Finslerian Gauss-Bonnet-Chern formulafor an even dimensional closed and oriented Finsler manifold (M,F ).

1 Motivation and Main ResultA brief review of Finsler geometryA biref review of the GBC Formula in Riem. geom.Generalizations in the Finsler setting beforeMain result: A Finslerian GBC formula

2 A Mathai-Quillen-type Formula

3 Sketch Proof of Main Theorem

4 A Special Case: Berwald Space


A brief review of Finsler geometry

M: C∞ manifold of dim. 2n; π : TM → M the tangent bundle of M.Given a local coordinate chart (U; (x i)) of M, there is acorresponding local coordinate chart (π−1(U), (x i , y i)) of TM.(M,F ) is called a Finsler manifold if F is a C∞ functionF : TM0 = TM \ 0→ R+ with the properties:(i) F (x , λy) = λF (x , y), ∀λ ∈ R+;(ii) (gij(x , y)) =

(12

[F 2]

y i y j

)> 0 on TM0.

So gij(x , y) are homogenous functions of degree zero on y , andso all geometric quantities coming from gij actually live on the unitsphere bundle SM of (M,F ).


Using (gij(x , y)) one can introduce a natural Euclidean structureon the pull-back bundle π∗TM → TM0.The Euclidean vector bundle (π∗TM,g) is essential in the study ofFinsler geormetry.There are two connections often used in Finsler geo.: one is theChern conn. ∇Ch, and the other is the Cartan conn. ∇Car.On the other hand, the Finsler metric F determines canonically ahorizontal subbundle H(TM0) of T (TM0) and the identification:

H(TM0) ∼= π∗TM.


let V (TM0) denote the natural vertical subbundle of T (TM0).

There is a canonical isomorphism J : H(TM0) ∼= V (TM0), by whichand the iso. H(TM0) ∼= π∗TM, the space TM0 admits awell-defined Riemannian metric gT (TM0).

As a natural foliated manifold (TM0,V (TM0)), it is well-known thatthere is a Bott connection ∇H(TM0) on H(TM0), which is flat alongthe leaves in TM0, that is(

∇H(TM0))2

(U,V ) = 0

for any U,V ∈ V (TM0).


Proposition 1 (F-Li, CAG, 2013)Under the identification H(TM0) ∼= π∗TM.

the Bott connection ∇H(TM0) on H(TM0) is exactly the Chernconnection ∇Ch on π∗TM;the symmetrization ∇H(TM0) of the Bott connection turns out to bethe Cartan connection ∇Car.

Remark 1Let RCh = (∇Ch)2 be the curvature of Chern connection. By theflatness along leaves, RCh splits only into two parts

RCh = R + P,

where R is an End(π∗TM)-valued horizontal two-form on TM0, and P isan End(π∗TM)-valued horizontal-vertical two-form on TM0.


A biref review of the GBC Formula in Riem. geom.

In 1943, S.S. Chern gave a simple and intrinsic proof of thefollowing Gauss-Bonnet-Chern formula for closed Riemannianmanifold (M,gTM) of even dimension:

Proposition 2 (S.S. Chern)

χ(M) =

(−12π

)n ∫M

Pf(RTM),

where RTM is the curvature of the Levi-Civita connection ∇TM

associated to the Riemannian metric gTM on M, and the PfaffianPf(RTM) is a well-defined 2n-form on M constructed by the curvatureRTM of ∇TM .


Chern’s remarkable transgression formula on SM(−12π

)n

π∗Pf(RTM) = −dSMΠ.

Let X ∈ Γ(TM) with the isolated zero point set Z (X ). Then onMε = M \ Zε(X ),(

−12π

)n

Pf(RTM) = −[X ]∗dSMΠ = −dM [X ]∗Π.

By using the Poincare-Hopf theorem, he obtained(−12π

)n ∫M

Pf(RTM) = limε→0

∫∂Zε(X)

[X ]∗Π = χ(M).


Generalizations in the Finsler setting before

Many people have made important contributions to thegeneralizations of the Chern’s formula in the Finsler setting:

Lichnerowicz, Bao-Chern, Z. Shen, Lackey, · · · .


Lichnerowicz’s Work(1949)

Lichnerowicz constructed an analogue differential 2n-formPf(RCar) on SM and proved a transgression formula

Pf(ΩCar) = −dSMΠCar,

for some 2n − 1-form ΠCar on SM, where RCar denotes thecurvature of the Cartan connection on π∗TM.Following Chern’s strategy, Lichnerowicz also made thecomputations∫

M[X ]∗Pf(ΩCar) := lim

ε→0

∫Mε

[X ]∗Pf(ΩCar) = limε→0

∫∂Zε(X)

[X ]∗ΠCar.


To get the desired Euler number χ(M) from the above computations,Lichnerowicz needed the following assumptions:

The space (M,F ) should be a Cartan-Berwald space.All Finsler unit spheres SxM = Y ∈ TxM|F (Y ) = 1 should havethe same volume as a Euclidean unit sphere

Note that the later assumption holds automatically for allCartan-Berwald spaces of dimension larger than 2.


Bao and Chern’s Work(Ann. of Math. 1996)

Around fifty years later, Bao and Chern dropped the Cartan-Berwaldcondition of Lichnerowicz by using the Chern connection ∇Ch.

They proved a transgression formula

Pf(ΩCh) + F = −dSMΠCh,

where Pf(RCh) is some 2n-form on SM constructed from thecurvature RCh of the Chern connection ∇Ch, F is a correction termand ΠCh is the associated transgression form.Then they established the following GBC-formula for all2n-dimensional oriented and closed Finsler manifolds with theconstant volume of Finsler unit spheres:∫

M[X ]∗[Pf(ΩCh) + F ] = χ(M)

Vol(FinslerSn−1)

Vol(Sn−1)


Shen and Lackey’s works

The constant volume condition in Bao-Chern’s formula is a muchstrong assumption for a Finsler manifold.To avoid this defect, following Bao-Chern’s approach, Z. Shen andLackey independently modified the GBC-integrand terms by usingthe unit sphere volume function V (x) = Vol(SxM) and obtainedsome new types of GBC-formulae respectively via the Cartan andthe Chern connections for all oriented and closed Finslermanifolds.Roughly, their formulae are of the form:

χ(M) = Vol(Sn−1)

∫M

[X ]∗[Pf(ΩCh) + F ]

Finsler(Sn−1)


Remark

A striking difference from the Chern’s formula for Riemannianmanifolds, all generalizations of the GBC formula in the Finslersetting before had to make use of an extra vector field X on M toobtain the related GBC-integrands.

In a recent paper of Prof. Y. Shen, he asked explicitly whetherthere is a GBC-type formula for general Finsler manifolds withoutusing any vector fields.

In the following, we try to answer Prof. Shen’s question by using aMathai-Quillen-type formula obtained by Weiping Zhang and myselfrecently.


A Finslerian GBC formula

Main theorem (F-Li)

Let (M,F ) be a closed and oriented Finsler manifold of dimension 2n.Then for any connection ∇ on π : TM → M, one has

χ(M) =1

(2π)2n(2n)!

n∑

k=1

C2k2n

∫SM

trs

[c(e)c(∇Che)2k−1(R\)k (P\)2n−2k

]+

∫SM

trs

[θ$

P$2n−1

],

where e = Y |SM and R\, P\, θ\ are the natural lifting of R, P, θrespectively on Λ∗(π∗T ∗M), and θ is defined by θ =: ∇Ch − π∗∇. Inparticular, the formula is independent of the choice of the connection ∇on TM.


A Mathai-Quillen-type Formula

In a recent work of Weiping Zhang and myself, we obtained aMathai-Quillen-type formula on the Euler number χ(M).More precisely, we applied the Mathai-Quillen’s geometricconstruction of the Thom class to the pull-back exterior algebrabundle π∗Λ∗(T ∗M)→ TM and obtained an integrable top-form onTM from any connection ∇ on TM, and proved that the integrationof this top form over TM is the Euler characteristic χ(M).Here the key point is that the connections used needn’tmetric-preserving!


In the following, we will recall briefly this Mathai-Quillen-type formula.

Let ∇TM be any connection on TM. Then it induces a connection∇Λ∗(T∗M) on the exterior algebra bundle Λ∗(T ∗M), whichpreserves the even/odd Z2-grading in Λ∗(T ∗M).Let Y denote the tautological section of the pull-back bundleπ∗TM:

Y (x ,Y ) := Y ∈ (π∗TM)|(x ,Y ),

where (x ,Y ) ∈ TM with x ∈ M and Y ∈ TxM.For any given Euclidean metric gTM on TM, let Y ∗ denote the dualof Y with respect to the pull-back metric π∗gTM on π∗TM.


Then the Clifford action c(Y ) = Y ∗ ∧ −iY acts on π∗Λ∗(T ∗M) andexchanges the even/odd grading in π∗Λ∗(T ∗M). Moreover,

c(Y )2 = −|Y |2π∗gTM = −|Y |2gTM .

For any T > 0, define the superconnection

AT = π∗∇Λ∗(T∗M) + Tc(Y )

on the bundle π∗Λ∗(T ∗M).

Then we have the following Mathai-Quillen-type formula:

Theorem 1 (F-Zhang, 2016)

χ(M) =

(1

2π

)2n ∫TM

trs[exp(A2T )].


A slight generalization of Theorem 1

One can choose any connection ∇ with the curvature R = ∇2 boundedalong fibres of TM and any Euclidean metric g on the pull-back bundleπ∗TM to define a superconnection on π∗Λ∗(T ∗M) ≡ Λ∗(π∗T ∗M).

Let ∇Λ∗(π∗T∗M) be the lifting of the connection ∇ on Λ∗(π∗T ∗M);Let Y ∗g denote the dual of Y with respect to the metric g and setcg(Y ) = Y ∗g ∧ −iY ;For any T > 0,

AT = ∇Λ∗(π∗T∗M) + Tcg(Y ) (1)

is also a superconnection on π∗Λ∗(T ∗M).


Corollary 1Let M be a closed and oriented manifold of dimension 2n. Then for anyconnection ∇ and any Euclidean metric g on π∗TM, if the curvatureR = ∇2 is bounded along fibres of TM, then the following formulaholds for any T > 0:

χ(M) =

(1

2π

)2n ∫TM

trs[exp(A2T )],

where AT is defined by (1).


Sketch Proof of Corollary 1

Set for any T > 0 and t ∈ [0,1],

ωT = AT − AT = ∇Λ∗(π∗T∗M) − π∗∇Λ∗(T∗M) + T (Y ∗g − Y ∗)∧,

AT ,t = tAT + (1− t)AT = π∗∇Λ∗(T∗M) + tωT + Tc(Y ).

Since the curvature R = ∇2 is bounded along fibres of TM, oneverifies that trs

[ωT exp(A2

T ,t )]

is exponentially decay along fibres

of π : TM → M, and so∫

TM/M

∫ 10 trs

[ωT exp(A2

T ,t )]

dt is awell-defined differential form on M.


Therefore, ∫TM

trs

[exp(A2

T )]−∫

TMtrs

[exp(A2

T )]

=

∫TM

dTM∫ 1

0trs

[ωT exp(A2

T ,t )]

dt

=

∫M

∫TM/M

dTM∫ 1

0trs

[ωT exp(A2

T ,t )]

dt

=

∫M

dM∫

TM/M

∫ 1

0trs

[ωT exp(A2

T ,t )]

dt

=0.


Sketch Proof of Main Theorem

In the following we will use Corollary 1 to establish a FinslerianGauss-Bonnet-Chern formula, in which no extra vector field isinvolved.In particular, an explicit GBC-type integrand on M will be given inthe induced homogeneous coordinate charts on SM.


Main theorem (F-Li)

Let (M,F ) be a closed and oriented Finsler manifold of dimension 2n.Then for any connection ∇ on π : TM → M, one has

χ(M) =1

(2π)2n(2n)!

n∑

k=1

C2k2n

∫SM

trs

[c(e)c(∇Che)2k−1(R\)k (P\)2n−2k

]+

∫SM

trs

[θ$

P$2n−1

],

where e = Y |SM and R\, P\, θ\ are the natural lifting of R, P, θrespectively on Λ∗(π∗T ∗M), and θ is defined by θ =: ∇Ch − π∗∇. Inparticular, the formula is independent of the choice of the connection ∇on TM.


Sketch Proof of Main Theorem

Step 1. Extended Chern connections

Let Dr M = (x ,Y ) ∈ TM|F (x ,Y ) < r be the r -disc bundle for∀r > 0.Let ρ ∈ C∞(TM) such that 0 ≤ ρ ≤ 1, and

ρ =

0, on TM \ D 12M;

1, on D 14M.

The extended Chern connection ∇Ch is defined on π∗TM → TM,

∇Chρ = (1− ρ)∇Ch + ρπ∗∇.

We similarly get a smooth metric g on π∗TM → TM from thefundamental tensor gF on TM0.


Step 2.For any T > 0, we define the following superconnection

Aρ,T = ∇Ch,\ρ + TcgF

(Y ).

Note that limT→+∞ exp(−T 2‖Y‖2) = 0 for F (Y ) ≥ 1, we have

χ(M) =

(1

2π

)2n ∫TM

trs

[exp(A2

ρ,T )]

= limT→+∞

(1

2π

)2n ∫D1M

trs

[exp(A2

ρ,T )].


Step 3.Note that trs

[exp(π∗∇Λ∗(T∗M))2] = 0, we have∫

D1Mtrs

[exp(A2

ρ,T )]

=

∫D1M

trs

[exp(A2

ρ,T )]−∫

D1Mtrs

[exp((∇Ch,\

ρ )2)]

+

∫D1M

trs

[exp((∇Ch,\

ρ )2)]−∫

D1Mtrs

[exp(π∗∇Λ∗(T∗M))2

].


Step 4.The first transgression:∫

D1Mtrs

[exp(A2

ρ,T )]−∫

D1Mtrs

[exp((∇Ch,\

ρ )2)]

=

∫SM

∫ 1

0i∗trs

[Tc(Y ) exp(∇Ch + Ttc(Y ))2

]dt

=n∑

k=1

C2k2n

(2n)!

∫SM

trs

[c(e)c(∇Che)2k−1(R\)k (P\)2n−2k

]


Step 5.Set

θρ = ∇Chρ − π∗∇.

The second transgression:∫D1M

trs

[exp((∇Ch,\

ρ )2)]−∫

D1Mtrs

[exp(π∗∇Λ∗(T∗M))2

]=

∫SM

∫ 1

0i∗trs

[θ\ρ exp(∇t )

2]

dt

=

∫SM

∫ 1

0trs

[θ\ exp

(RCh,\ − (1− t)[∇Ch,\, θ\] + (1− t)2θ\ ∧ θ\

)]dt .


Note that

[∇Ch,\, θ\] = θ\ ∧ θ\ −(π∗∇Λ∗(T∗M)

)2+ RCh,\.

The term θ\ is an End(Λ∗(π∗T ∗M))-valued horizontal one form,and so

tR\ + (1− t)(π∗∇Λ∗(T∗M))2 − t(1− t)θ\ ∧ θ\

is an End(Λ∗(π∗T ∗M))-valued horizontal two form.


∫D1M

(trs

[exp((∇Ch,\

ρ )2)]− trs

[exp((π∗∇Λ∗(T∗M))2)

])=

∫SM

∫ 1

0trs

[θ\ exp

(tP\+tR\+(1− t)(π∗∇Λ∗(T∗M))2 − t(1− t)θ\ ∧ θ\

)]dt

=

∫SM

∫ 1

0trs

[θ\ exp

(tP\)]

dt

=

∫SM

1(2n − 1)!

∫ 1

0t2n−1dt trs

[θ$P$2n−1

]=

1(2n)!

∫SM

trs

[θ$

P$2n−1

].


In the homogeneous coordinate charts (x i , y i) on SM, we have:

Corollary 2 (F-Li)

Let (M,F ) be a closed and oriented Finsler manifold of dimension 2n.Let RCh = R + P be the curvature of the Chern connection ∇Ch on thepull-back bundle π∗TM over SM. Then in the induced homogeneouscoordinate charts (x i , y i) on SM, one has

χ(M) =1

(2π)2n(2n)!

n∑

k=1

(−1)kC2k2n Ck−1

2k−2

∫SM

δi1...i2nj1...j2n

R j1i1· · ·R jk

ik

P jk+1ik+1· · ·P j2n−k

i2n−k·Υj2n−k+1

i2n−k+1· · ·Υj2n−1

i2n−1Ξj2n

i2n

+

∫M

∫SM/M

δi1...i2nj1...j2n

P j1i1· · ·P j2n−1

i2n−1$j2n

i2n

,

where $ji , R j

i , P ji , Υj

i and Ξji are defined respectively in the following:


($ji ) is the connection matrix of the Chern connection ∇Ch with

respect to natural frames on π∗TM;(R j

i ), (P ji ) are the curvature matrices of the R-part, P-part of the

curvature (∇Ch)2, respectively;

(∇Che

)i:= d

(y i

F

)+

y j

F$i

j , (∇Ch,∗ω)i := dFyi − Fy j$ji ,

Υji := (∇Ch,∗ω)i

(∇Che

)j;

Ξji :=

(Fy i (∇Che)j − y j

F(∇Ch,∗ω)i

);

e = Y/F (Y ) and ω = e∗ is the Hilbert form.


Sketch Proof of Corollary 2

Let (x i , y i) be any homogeneous coordinate charts on SM.Set

∇Ch ∂

∂x j = $ij ⊗

∂

∂x i , ∇ ∂

∂x j = ϑij ⊗

∂

∂x i

Set

R∂

∂x j = R ij ⊗

∂

∂x i , P∂

∂x j = P ij ⊗

∂

∂x i .


We have

$\ = −$ji dx i ∧ i ∂

∂x j, ϑ\ = −ϑj

idx i ∧ i ∂∂x j

R\ = −R ji dx i ∧ i ∂

∂x j, P\ = −P j

i dx i ∧ i ∂∂x j.

In particular,

P ij = P i

j kldxk ∧ δyl

F= −dxk ∧

(∂Γi

jk

∂y l δyl

).


Denote that

c(e) = ω ∧ −ie = Fy j dx j ∧ −y i

Fi ∂∂x i,

where ω = e∗ = Fy i dx i is the Hilbert form on SM;

(∇Che

)i:= d

(y i

F

)+

y j

F$i

j , (∇Ch,∗ω)i := dFyi − Fy j$ji ,

Υji := (∇Ch,∗ω)i

(∇Che

)j, Ξj

i :=

(Fy i (∇Che)j − y j

F(∇Ch,∗ω)i

).


We have

trs

[c(e)c(∇Che)2k−1(R\)k (P\)2n−2k

]=trs

[(−R j

i dx i ∧ i ∂∂x j

)k (−P j

i dx i ∧ i ∂∂x j

)2n−2k

((∇Ch,∗ω)jdx j ∧ −(∇Che)i i ∂

∂x i

)2k−2

((∇Ch,∗ω)pdxp ∧ −(∇Che)q i ∂

∂xq

)(Fy l dx l ∧ −y r

Fi ∂∂x r

)]=(−1)kCk−1

2k−2δi1...i2nj1...j2n

R j1i1· · ·R jk

ikP jk+1

ik+1· · ·P j2n−k

i2n−kΥ

j2n−k+1i2n−k+1

· · ·Υj2n−1i2n−1

Ξj2ni2n.


trs

[θ$

P$2n−1

]=trs

[−($j

i − π∗ϑj

i

)dx i ∧ i ∂

∂x j

(−P l

kdxk ∧ i ∂∂x l

)2n−1]

=δi1...i2nj1...j2n

P j1i1· · ·P j2n−1

i2n−1

($j2n

i2n− π∗ϑj2n

i2n

).

Note that P ji are vertical exact, we get∫

SM/MP j1

i1· · ·P j2n−1

i2n−1(π∗ϑj2n

i2n) = ϑj2n

i2n

∫SM/M

P j1i1· · ·P j2n−1

i2n−1= 0.


Therefore, we get

1(2n)!

∫SM

trs

[θ$

P$2n−1

]=

1(2n)!

∫M

∫SM/M

trs

[θ$

P$2n−1

]=

1(2n)!

∫M

∫SM/M

δi1...i2nj1...j2n

P j1i1· · ·P j2n−1

i2n−1$j2n

i2n


Berwald spaces

The P-part of the Chern curvature vanishes for Berwald spaces.With respect to any orthonormal frame e1, · · · ,e2n of π∗TM withe2n = e, set

∇Chei = ωji ej , RChei = Rei = R j

i ej ,

and so

RCh,\ = −R ji e∗,i ∧ iej = −1

4R j

i (c(ei) + c(ei))(c(ej)− c(ej)).


Corollary 3Let (M,F ) be a closed and oriented Berwald space. Then one has,

χ(M) =

(−12π

)n 1Vol(Sn−1)

∫M

∫SM/M

Pf(RCh)ω2n1 · · ·ω2n

2n−1,

where R ji =

(R ji−R i

j )

2 and

Pf(RCh) =1

2nn!

2n∑i1,...,i2n=1

εi1,...,i2nR i2i1∧ · · · ∧ R i2n

i2n−1.


Berwald surfaces

Combining with Bao-Chern’s result for closed and oriented Berwaldsurfaces: ∫

M−R 2

1 12ω1 ∧ ω2 = χ(M)Vol(FinslerS1),

we get

Corollary 4

Let (M,F ) be a closed and oriented Berwald surface, then[Vol(FinslerS1)− 2π

]χ(M) = 0, (2)

that is, Vol(FinslerS1) = 2π, or χ(M) 6= 0.


In fact, by the Szabo’s rigidity theorem that any Berwald surfaces mustbe locally Minkowskian or Riemannian, and a closed locallyMinkowskian surface has zero Euler number, one also gets (2) easily.


Thank You!


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