Introduction to Finsler Geometry

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Peking University tfzSjE Series in Mathematics — Vol. 1

AN INTRODUCTION TO

FINSLER GEOMETRY Xlaohuan Mo

AN INTRODUCTION TO

FINSLER GEOMETRY

PEKING UNIVERSITY SERIES IN

MATHEMATICS

Series Editor: Kung-Ching Chang (Peking University, China)

Vol. 1: An Introduction to Finsler Geometry by Xiaohuan Mo (Peking University, China)

Peking University (lfc?sP Series in Mathematics — Vol. 1

"W-IJ,

"'••;:ix.

AN INTRODUCTION TO

FINSLER GEOMETRY Xiaohuan Mo

\jj^ World Scientific NEW JERSEY

• LONDON

• SINGAPORE

• BEIJING

• SHANGHAI

• HONG KONG

• TAIPEI

• CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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AN INTRODUCTION TO FINSLER GEOMETRY Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-256-793-3

Printed in Singapore by World Scientific Printers (S) Pte Ltd

Preface

This book uses the moving frame as a tool and develops Finsler geometry on the basis of the discovery of Chern connection and projective sphere bundle. It systematically introduces three classes of geometrical invariants on Finsler manifolds and their intrinsic relation, analyses local and global results from classic and modern Finsler geometry, and gives non-trivial examples of Finsler manifolds satisfying kinds of curvature conditions. Most of the geometric invariants and fundamental equations in this book lie on the projective sphere bundle, that is, they are positively homogenous degree zero with respect to the tangent vectors. The content of this book is brief and to the point. We reveal the studying method of modern Fisher geometry by introducing the theory of harmonic maps of a Fisher space and the geometry of a projective sphere bundle. This book is easy to understand. The author has given a one-semester course Finsler Geometry three times to graduate students of mathematics of Peking University. The book is based on the teaching of these courses. S. S. Chern, a renowned leader in differential geometry, provided me with generous support and guidance for my development of this manuscript and preparation for my teaching. I would like to take this opportunity to express my sincere gratitude to him. I am also grateful to C. Yang and L. Huang for generous help. I would like to express my deep gratitude to Z. Shen numerous suggestions which led to significant improvements of both the style and content of the book. X. H. Mo

v

This page is intentionally left blank

Contents

Preface

v

1.

Finsler Manifolds

1

1.1 1.2 1.3 1.4 1.5

1 2 4 6 9

2.

3.

4.

Historical remarks Finsler manifolds Basic examples Fundamental invariants Reversible Finsler structures

Geometric Quantities on a Minkowski Space

11

2.1 2.2 2.3 2.4 2.5

11 12 14 15 17

The Cartan tensor The Cartan form and Deicke's Theorem Distortion Finsler submanifolds Imbedding problem of submanifolds

Chern Connection

23

3.1 3.2 3.3 3.4

23 26 32 37

The adapted frame on a Finsler bundle Construction of Chern connection Properties of Chern connection Horizontal and vertical subbundles of SM

Covariant Differentiation and Second Class of Geometric Invariants

39

4.1 Horizontal and vertical covariant derivatives 4.2 The covariant derivative along geodesic

39 40

vii

viii

5.

6.

7.

8.

9.

An Introduction

to Finsler

Geometry

4.3 Landsberg curvature 4.4 5-curvature

42 44

Riemann Invariants and Variations of Arc Length

53

5.1 5.2 5.3 5.4

Curvatures of Chern connection Flag curvature The first variation of arc length The second variation of arc length

53 58 59 64

Geometry of Projective Sphere Bundle

69

6.1 Riemannian connection and curvature of projective sphere bundle 6.2 Integrable condition of Finsler bundle 6.3 Minimal condition of Finsler bundle

69 75 77

Relation among Three Classes of Invariants

81

7.1 7.2 7.3 7.4

81 83 85 85

The relation between Cartan tensor and flag curvature . . Ricci identities The relation between 5-curvature and flag curvature . . . . Finsler manifolds with constant S'-curvature

Finsler Manifolds with Scalar Curvature

89

8.1 Finsler manifolds with isotropic S'-curvature 8.2 Fundamental equation on Finsler manifolds with scalar curvature 8.3 Finsler metrics with relatively isotropic mean Landsberg curvature

89

Harmonic Maps from Finsler Manifolds

99

9.1 9.2 9.3 9.4 9.5

Some definitions and lemmas The first variation Composition properties The stress-energy tensor Harmonicity of the identity map

91 95

99 102 108 112 114

Bibliography

117

Index

119

Chapter 1

Finsler Manifolds

Finsler geometry is just Riemannian geometry without the quadratic restriction. It also asserts itself in applications, most notably in theorey of relativity, control theory and mathematical biology. In this chapter, we will introduce Finsler manifolds and discuss the relation between Finsler manifolds and Riemannian manifolds. Also, we will give other useful examples of Finsler manifolds [AIM, 1993].

1.1

Historical remarks

The geometry based on the element of arc ds = F(x\---

,xn;dx\---

,dxn)

(1.1)

where F is positively homogeneous of degree 1 in dxz is now called Riemannian-Finsler geometry (Finsler geometry for short). Roughly speaking, F is a collection of Minkowskian norms Fx in the tangent space at x such that Fx varies smoothly in x. In fact, metric (1.1) was introduced by Riemann in his famous 1854 Habilitationsvortag "Uber die Hypotheser welche der geometric zugrund liegen". He paid particular attention to a metric denned by the positive square root of a positive definite quadratic differential form, i.e. F2(x, dx) =

gij(x)dxldxj

In the famous Paris address of 1900, Hillbert devoted the last problem to variational calculous of f ds and its geometrical overtone. A few years later, the general development took a curious turn away from the basic aspects and methods of the theory as developed by P. Finsler. The latter 1

An Introduction

2

to Finsler

Geometry

did not make use of the tensor calculous, being guided in principle by the notions of the calculous of variations. Finsler's thesis, which treats curves and surfaces of (1.1), must be regarded as the first step in this direction. The name "Finsler geometry" came from his thesis in 1918.

1.2

Finsler manifolds

Let M be an n-dimensional smooth manifold. Denote by TXM the tangent space at x G M, and by TM — \JxcM T^M the tangent bundle of M. Each element of TM has the form (x,y), where x G M and y G TXM. The natural projection 7r : TM —» M is given by TT(X, y) = x.

M

x

Fig. 1.1

Let (tp,xl) be a local coordinate system on an subset U C M. i.e. VrrGt/,

,xn).

Taking a curve on U li(t)

=

1

lp-\x

,-

, x% + t, xi+l

,*"),

then gfrlx = 7;(0) are the induced coordinate bases for TXM. We said xl gives rise to local coordinates (xl,yl) on 7r_1f/ C TM through the mechanism d y = y dxl

(1.2)

The yl are fibrewise global. Whenever possible, let us make no distinction between (x,y) and its coordinate representation {xl,yl). Function H that defined on TM can be locally expressed as H(x\...,xn-y1,..-,yn).

Finsler

3

Manifolds

We employed the following convention, namely, denoted by Hyi, Hyiyj, • • • , etc, the partial derivative(s) of H with respect to the coordinate y%. Adopt a similar notation for the partial derivative with respect to the coordinate x\ Lemma 1.2.1 Let V be an open subset on M, such that V C\U ^ O, Let (xl,yl) be the local coordinate on •K~1V C TM. Then (i)

(«)

Proof.

tf,

H

#

=

(L3)

WH"i-

= _ ^ ,

^ ^

+

v

, .

(1.4)

By (1.2) we have dxi

•

(L5)

%

y = w*This implies that ^

= ~ .

From (1.5) and (1.6) and the chain rule, we have (1.3) and (1.4).

(1.6) D

Definition 1.2.2 A function F : TM —> [0, oo) is called a Finsler structure, if, in a local coordinate systerm (xl,yl),

F

^y) =

F

{y^)

satisfies (i) F(x,Xy) = XF{x,y); A > 0. (ii) F(x, y) is C°° for y ^ 0. (iii) \{F2)yiyi (y i1 0) is positive definite. A C°° manifold M with its Finsler structure F is said a Finsler manifold or Finsler space.

An Introduction

4

1.3

to Finsler

Geometry

Basic examples

(I) Riemannian manifold A Riemannian metric g on a manifold M is a family of inner products {gx}X£M s u c n that the quantities f \

( d

d

are smooth in local coordinates. The Finsler structure F(x, y) of a Riemannian manifold satisfies

F(x,y) = \Jgij{x)yiyK In that case, h(F2)yiyj is simply gij{x), which is independent of y. Let | . | and (,) be the standard Euclidean norm and inner product in Rn. We have the following interesting Finsler metrics defined on Bn

\y\2-{\xW-(x,y?)^

F=

r _yi3/i

2

(L6)

-(M%i 2 -(^) 2 \

(L7)

For metric (1.6), we have 9ij

i

22 '

yivi

rsWw(i-M2) + (£fc*V)2 1-1 2

y^ 3 '

i

<5jj(l-|x| )+x ^

= gij{x).

So metric (1.6) is a Riemannian metric. Similarly we can show that the metric defined in (1.7) is also Riemannian. These Riemannian metrics have special properties. Riemannian metric in (1.6) has constant curvature 1, and this in (1.7) has constant curvature -1. The notion of curvature will be defined later. The metric in (1.7) is called the Klein metric. (II) Minkowski manifold

Finsler

Manifolds

5

Definition 1.3.1 Let (M,F) be a Finsler manifold, we say that (M,F) is a Locally Minkowski manifold if for arbitrary chart (U,xz) of M, the fundamental tensor satisfies gij(x, y) = gij(y). Furthermore, we call (M, F) a (globally) Minkowski manifold if M is a vector space. E x a m p l e 1.3.2 Consider the case n = 2, so y = (y 1 ,?/ 2 )- In order to avoid the excessive use of parenthese, we shall abbreviate y1 ,y2 as p,q respectively. Define Fx,k = \/p2 + q2 + X(p2k +

q2k)i,

where A E [0, oo), k e {1,2, • • • } . The case A = 0 is Riemannian (in fact, Euclidean) [Bao and Shen, 2002]. It can be checked that this F is actually C°° on TM\{0}, calculations give 1 2

d2Fjk dp2 i

where

Hence

Q2

1 + Awp 2 ( fc - 1 V fc + (2* - l)<72fc] > 0, pi

2-W=2A(1^MP# uj = (p2k + I' d2F2

q2k)i'2. \

One obtains a Minkowski structure. (Ill) Randers manifold Definition 1.3.3 Let a = y/aij (x) be a Riemannian metric on a differential manifold M and /3 = bi(x)yl be a 1-form on M. Suppose that

\\P\\a = yJaVbib'j < 1, where (a1^) = ( a j j ) - 1 . Define

F = a + f3

6

An Introduction

to Finsler

Geometry

F is a (positively definite) Finsler metric. F is called the Randers metric. It was first studied by physicist G. Randers, in 1941 from the standard point of general relatively [Randers, 1941]. Example 1.3.4 The following Randers metrics are very special. It is the deformation of Klein metric in (1.7). F

_ V\y\2 - (N 2 |y| 2 - (x,y)2) l-W2 x£Bn,

(x,y) l-ld2

(

y e TxBn = TxRn.

It is easy to show

is an exact form. In particular, (3 is closed; (n)

||/3||a
The pair of metrics in (1.9) are called the Funk metrics on the unit ball. Example 1.3.5

Consider =

(l-e2)(x,y)2

+ e\y\2(l

a

+ e\x\2)

1 + eld

2

\/l-e2(x,y) P

1

i

I

19

z

1 + e\x\

where x £ R2,y

€ TXR2, e G (0,1), then

Hence F = a + (3 is a Randers metric.

1.4

Fundamental invariants

(I) Fundamental tensor

•

'

Finsler

Manifolds

7

L e m m a 1.4.1 Let V be a vector space and H : V —> R be homogeneous of degree r. That is, H{Xy) = XrH(y)

for

all

A > 0.

positively

(1.10)

Then yiHyi(y)=rH(y),

(1.11)

where (y 1 , • • • , yn) is the coordinate of y £ V, and H

y* -

dy^-

Proof. Suppose H satisfies (1.10). Fix y. Differentiating this equation with respect t o A gives yiHyi(Xy)=r\r-1H(y). Setting A equal t o 1 gives (1.11).

C o r o l l a r y 1.4.2

Let (M,F)

•

be a Finsler manifold.

Then

y'Fyi = F,

(1.12)

^ Fyiyj = 0,

(1.13)

ylFyiyjyk

= —Fyiyj.

(LH)

Proof. Succesively substituting F,Fyi,Fyiyj for H. Note t h a t F,Fyi, Fyiyj is positively homogeneous of degree 1,0, and — 1, respectively.

•

Let (M, F) be a Finsler manifold. Setting g : = gij(x,y)dxl

®dx3,

where 9ij

:=

n{F ) y V = FFyiyj

+ FyiFyj.

(1.15)

S

An Introduction

to Finsler

Geometry

It is easy to show that the symmetric covariant 2-tensor g is intrinsically defined on TM by using Lemma 1.2.1. We call g the fundamental tensor of F . Lemma 1.4.3 The components of the fundamental following properties:

tensor satisfy the

(i)

y'gy = FFyi,

(1.16)

(")

yVffy = ^ 2 ,

(1.17)

(«*) » 9j/k ^ =^

<9yk

= y *<9yk ^ = o.

(1.18)

Proof, (i) and (ii) follows from (1.12), (1.13) and (1.15). (iii) Substituting gtj for H in Lemma 1.4.1. Note that gij is positively homogeneous of degree 0. • (II) Hilbert form Let M be a smooth manifold and SM the projective sphere bundle of M, with canonical projection map p : SM —> M given by (x, [y]) —> a; and 5 X M := p _ 1 (x) « 5 " - 1 , the projective sphere at x.

M

x

Fig. 1.2

Let (M,F) be a Finsler space and (xl,yl) the local coordinate on T M . Setting u> := Fyjdx^, then we have the following Lemma 1.4.4

u> is a globally defined differential form on SM.

Finsler

Manifolds

9

Proof. Taking another local coordinate on TM (cf. Lemma 1.2.1). From (1.2) we have Fyj di? = Fyj

dxj.

Thus u is globally defined. On the other hand, it is easy to see Fyi(x,Xy)=FyJ(x,y),

VA G R+.

This concludes that w is defined on SM.

•

Definition 1.4.5 Let (M, F) be a Finsler manifold . We call u> the Hilbert form of (M, F). Let (M, F) be an n-dimensional Finsler manifold and p : SM —> M the canonical projection map. We pull back TM and T*M, and denote them by p*TM and p*T*M respectively, making them vector bundles (with ndimensional fibres ) over the (2n — l)-dimensional base .SM. We call p*TM (resp. p*T*M) the Finsler bundle (resp. the duaZ Finsler bundle) [Chern, 1992].

M

SM Fig. 1.3

It is easy to see that the Hilbert form is a section of p*T*M.

1.5

Reversible Finsler s t r u c t u r e s

The Finsler structure in Example 1.3.2 has the following important property Fx,k(y) =

Fx,k(-y)-

10

An Introduction

to Finsler

Geometry

In general, a Finsler structure F : TM —> R is said to be reversible if F(x,y)=F(x,~y),

(1.19)

for all (a;, y) £ TM. It is easy to show that a Finsler structure is reversible if and only if F(x,Xy) = \X\F(x,y),

(1.20)

for all A ^ 0. Of importance to our treatment with respect to reversible Finsler structure is the projectivized tangent bundle PTM obtained from TM by identifying its non-zero vectors which differ from each other by a multiplicative factor. Geometrically PTM is the manifold of the line elements of M. Let (M, F) be a reversible Finsler manifold. Since Fyi is homogeneous of degree zero in y, the Hilbert form lives in PTM. In classical calculus of variations it is called Hilbert's invariant integral. Similarly, the fundermental tensor in Section 1.4 also lives in PTM. Lemma 1.5.1 Let (M,F) be a Randers space. If F is reversible, then (M, F) is a Riemann manifold . Proof.

Consider a Randers metric F = a -f- /3, where a = yaij(x)yiy^;

(3 = bi(x)yl

Thus F(x,-y)

= y / a i j (x)(- 2 /')(-2/ j ) + M 0 ( - 2 / ) = \Jaij{x)yiyj

-bi{x)y\

Since F is reversible, F(x, ~y) = F(x, y). This implies that bi(x)yi = 0 = 0. We obtain F = a is Riemannian. •

Chapter 2

Geometric Quantities on a Minkowski Space

There are several geometric invariants on a Finsler manifold. Some of them vanish on a Riemannian manifold. So we call them non-Riemannian quantities. In this chapter, we introduce the first class of geometric invariants. These invariants describes the non-Euclidean properties of Minkowski tangent spaces over a Finsler manifold. 2.1

The Cartan tensor

Let (M, F) be a Finsler manifold with its fundermental tensor g^ (cf. Section 1.4). Put _ Fdgij

A := Aijkdxi

_ F

2

dxj ® dxk.

It is easy to see that A is well denned on SM. We call A the Cartan tensor of (M, F). Proposition 2.1.1 Let (M,F) be a Finsler manifold. Then (?) Aijk is totally symmetric, i.e. Aijk = Ajik = Aikj; (ii) y^Aijk = 0; (Hi) (M, F) is Riemannian if and only if A = 0. Proof. (i) and (Hi) follow from (2.1). (ii) Follows (1.18).

11

•

12

An Introduction

to Finsler

Geometry

Remark The Cartan tensor gives a measure of the failure of (M, F) to be a Riemannian manifold. Put g := (g^) and (gij) := {gij)-1. It is easy to see that ^ l o g V ^ = f ^ l o

g

d e t ,

=

f ^ ^ G , ,

(2.2)

where G^ is the algebraic complement of g^. It follows that Gij = (det g)gji = (det g)gij.

(2.3)

Plugging it into (2.2) and using (2.1) yields

Then we have following Lemma 2.1.2 Let (M,F) be a Finsler space. Then detg is independent of y if and only if Ai = 0 where Ai:=gjkAijk.

2.2

(2.4)

The Cartan form and Deicke's Theorem

Setting f] := Aidx1 where Ai is defined in (2.4). Then 77 is a well-defined 1-form on SM, and r) is called Cartan form. Let F : Rn -> [0,oo) be a Minkowski norm on Rn. Thus (i) F is positively y-homogeneous of degree one; (ii) F is C°° on i?"\{0} and the following symmetric matrix 9ij

'• =

2

y'y3

is positive definite. Put g := (g^) and (gi:>(x)) =

(gij(x))~1.

Lemma 2.2.1 For any x 6 Rn\{0}, define <j>x : Rn\{0} trace[g~1(x)g(y)}. If det g = const, then min

yeRn\{o}

<j>x{y) = 4>x(x)-

-+ R by <j>x(y) =

Geometric

Proof.

Quantities

on a Minkowski

Denote the eigenvalues of g

Space

(x)g(y) by Ai, • • • , An. Then

4>x(y) = A H — + An > n(AiA 2 ---A„)i = n{det[g-\x)e(y)]}^ = n{{detg(y)]-1{detg(y)}}i = n (since det g(y) = const) = tr[g-1(x)g(x)] =<j>x(x). Lemma 2.2.2 we have

13

Q

Consider the following operator A = g' J (x) 9% j • Then __ (

d^x \dyhdyk

Q

Proof. A 9hk = 9°ziz(x)\

O 9hk

dyi-dyi

dyhdykdyidy:> 9

9

[X)

dyidyidyhdyk

Hi

[X,

\_^9ij_

dyhdyk _ d^(x)gij(y)} QyhQyk

_ d*4>x dyhdyk'

rj

Lemma 2.2.3 Let F : Rn —> [0,oo) be a Minkowski norm. Assume that det g = constant then for all i,j, gij = constant. Proof. Setting S := {x e Rn\{0} | F(x) = 1}. Then S is compact. It follows that for any fixed fte{l,2,---,n}, there exist XQ e S, such that ghh(xo)=maxghh(x) xGS

=

max

ghh(x)

(2.5)

xGi?"\{0}

here we have used the fact that ghh is positively y-homogeneous of degree zero. By using Lemma 2.2.1 and the principle of calculus, we have ( d'ig*j ) {x) is semi-positive. Together with Lemma 2.2.2 we obtain Aghh > 0. Notice that (2.5) and Hopf's maximal principle we have

An Introduction

14

to Finsler

Geometry

ghh = constant. Thus Aghh = 0. Since Ag is semi-positive, Aghk = 0 for all h, k. A similar discussion implies that ghk — constant for h ^ k. •

Theorem 2.2.4([Deicke,1953]) Let (M', F) be a Finsler manifold. Then it is Riemannian if and only if its Cartan form vanishs identically.

Proof.

From Lemma 2.1.2 and Lemma 2.2.3.

•

R e m a r k The Cartan form characterizes the Riemannian manifolds among Finsler manifolds.

2.3

Distortion

Let Bn denote the unit ball in Rn. Let (M, F) be a Finsler manifold. Let

T

= In

y/det(gi:j(x,y))
where o-p(x) := ,. .. „„,"„,—^-3—T-TTT- T is called the distortion [Shen, 2001; Shen, 1997]. It is easy to see r is positively y-homogeneous of degree zero. Observe that dr _ d dyk dyk d ~

y/det(gi:j(x,y)) aF(x) d\naF{x)

ln det

dyk

\/ (9ij(^y))

_ \ ydgn _ 1 " 29 dy* ~ FAl3k9

(2.6)

dyk

ti_

1 ~

FAk-

(2.6) gives a relation between distortion and Cartan form. By Theorem 2.2.4 , we have Proposition 2.3.1 Let (M,F) be a Finsler manifold. Then (M,F) Riemannian manifold if and only if its distortion is defined on M.

is a

Geometric

2.4

Quantities

on a Minkowski

Space

15

Finsler submanifolds

Let (M, F) be a Finsler manifold and M a smooth manifold. Let / : M —> M be an immersion. Define /„ : TM —• T M by /.(*,!/) = (/(*), (#)*(*>)).

(2-7)

where (df)x : T X M -> Ty ( x ) M is the differentiation of / at x <E M. P u t F := F o /„. It is easy to show that F is a Finsler structure on M. F is called the Finsler structure induced by F. Lemma 2.4.1 Let f : M —> (M,F) be an immersion into a Finsler manifold (M,F) and F the Finsler structure induced by F. Then the fundamental tensor g and Cartan tensor A of F satisfies 9ab(x,y)=gi:,{x,y)

Aabc{^y)

— -^,

= Aljk{x,y)

— —b—^

(2.8)

(2.9)

where f% is the component function of f with respect to xl, xa is the local coordinate on M, and x:=f(x),

Proof.

y = {df)x{y)-

(2-10)

By (2.7),(2.10) and F = F o /„, we have F(x,y) = F(x,y).

(2.11)

Thus Fya(x,y)

dxl = Fi,(x,y)—+Fy,(x,y)-^

dff

~ dp = Fyi{x,y)-±-a{x),

(2.12)

where we have used (2.10) and

dy^=8f_ dya

dxa

Similarly, Fyayb(x,y) = F^(x,y)^(x)^(x).

(2.13)

An Introduction

16

to Finsler

Geometry

Now (2.8) is an immediate conclusion of (2.11)-(2.13). From (2.1) (2.8) and (2.11), we have Aabc{x,y)

=

-F(x,y)—gab(x,y) F{x,y)

8f_ Qp op M s . J / ) ^ ( a ) ^ ( s ) dyc df\ ,dfa-k&fav) dxdydxb dxc

8 dy-

F(x,y)

T^te^fc^

Hence we get (2.9).

•

Corollary 2.4.5 Let (M,F) be a Riemannian manifold. Then its induced Finsler structure is Riemannian. Let V be an n-dimensional vector space and / a Minkowski norm on V. Fix a basis {e;} on V. Then y = y%ei for y £V. Thus (y,u) has its local coordinate (y1,--- ,yn; u1,--- ,un), where u% — dyl{u), and (y,u) £ TV. Define F : TV -* [0, +oo) by F(y,u) = f(uiei).

(2.14)

Then it is easy to show that (i) F is well denned, i.e. F is independent of the choose of {ei} ; (ii) F is a Finsler structure on the linear manifold V; {m)F(y,u) = F(u). Put S := {y e V\F(y) = 1}. Denote the natural imbedding from S into V by h. By (2.11) and (2.14), the induced Finsler structure by F satisfies that F{u,w*7fc)

=

F{h{u),h.{w°£F))

= F(h(u),u;°&1fa)

(2.15)

For instance, consider Example 1.3.2 and e\ = (1,0), from (2.15)

e 2 — (0,1). Then

Geometric

2.5

Quantities

on a Minkowski

Space

17

Imbedding problem of submanifolds

Recall well-known Nash Theorem: Let M be an n-dimensional Riemannian manifold. Then there exist an isometric imbedding M S-> Rm where

m — '

^ - T J2— - ,

if

M

is

compact;

2(2n + l)(3n + 8),

if

M

is

noncompact,

Consider a Finsler manifold (M,F), its tangent space on each point is Minkowski. It is natural to ask the following: Problem If any n-dimensional Finsler space can be isometrically imbedded into a Minkowski space ? Let (M, F) be a Finsler manifold and A the Cartan tensor of F. Write Ix •= {y G TxM\F{y)

= 1}

and \A\\X :— sup sup

\AXiy(u,u,u)

y€lxu£lx

We call

||J4|| X

\gx,y(u,u)\2

the norm of A at x. In fact I,,,, \\A\\X :=

sup

sup

\AXty(u,u,u)\ — j-.

1

J/GT I M\{0} ueTxM\{0}

Proposition 2.5.1 Let f : (M,F) S-» (M,F) Then for any x € M, we have

\9x,y(u,u)\2

be an isometric

< WMfM, where A and A are the Cartan tensor of F and F respectively.

immersion.

(2-16)

18

An Introduction

to Finsler

Geometry

Proof. |, ,.11

\Ax,y(u,U,u)\

\\A\\X : = sup sup i—-*V£la:U£lx

-I-

\gXiy(U,U)\2

IA/(z),/.(y))(/* u >/*">/*")I 3—

= sup sup ——^—•^- LL 2/e/* «£/»

\9(f(x),f,(y))(f*U,

s <

f*U)\2

U(/(»),/. (»))("»"•") I

sup

sup

ye//(x) « / / ( j )

— 1 - L - L ! —^^

3—

|g(/(rc),jj)(u,M)|2

Corolary 2.5.2 / / a Finsler manifold can be embedded into a finitedimensional Minkowski manifold, then the Cartan tensor must be bounded. Proof. Let V be a finite dimentional vector space and FQ a Minko wski norm on V. By Section 2.4, we define a Finsler structure F from FQ such that F(x,y) = F0(y).

(2.17)

Let / : (M,F) >—> (V, F) be an isometric immersion. From (2.17) we have \\A\\X =

constant.

Together with Proposition 2.5.1, sup \\A\\X < sup ll-AH^a;) = constant < +oo. xgM

x£M

|-I

Consider the following Minkowski manifold (Rn,F) F = Fx(y) = yf^ly*]*

+ \r*

where

(A > 0),

r=0>>T)}.

(2.18)

(2.19)

Notice that when n = 2, then F \ = F\,2 (see Example 1.3.2). Now

F2 = ^ > f + Ar2, r4

= 5>i4.

( 2 - 2 °)

( 2 - 21 )

Geometric Quantities

on a Minkowski

Space

19

By (2.21) we get r

V

= r-V)3.

(2.22)

Thus (r-2),. = - 2 r - V ) 3 , (r-%

(2.23)

= -6r"10(^)3,

(2.24)

=yi + Xr-2(yi)3.

(2.25)

From (2.20) and (2.22), we have ^-)

Together with (2.22) and (2.23), we get = Stj + 3 A r - W < % - 2Ar- 6 (j/V') 3 -

9ij(y) = (~)

(2-26)

A simple calculation give the following formula (yVSi^y*

= 2y%jk,

(2.27)

where Sijk = \l' 10,

* ' = other.

j

= k

''

(2-28)

+ 2/W)3<^]-

(2-29)

Similarly Wy%

= 3[yW)38ik

Thus, from (2.23),(2.24),(2.26),(2.27) and (2.29)

= tflZr-^Sij = 3\F[r-2yi6ijk

-r-ttfvWH*

- 2r-6(yiy^]yk + 2r-10(yiyJyk)3

+ yiy^y'ftjk

. V

. '

J

+ 2/V (yfc)3%)]-

In particular, from (2.26) and (2.30), we get gu(y) = 1 + 3 A r " V | 2 - 2Ar~ V | 6 , A m ( y ) = 3 A F [ r ~ y + 2r~w{ylf

- Zr^^f).

(2-31) (2.32)

20

An Introduction

to Finsler

Geometry

Taking !/o = (l,A*,0,--- ,0),

(2.33)

r(y 0 ) = (l + A 2 )i.

(2.34)

then

Together with (2.31) and (2.33), 1

flnM

l Q \2

= l + A ; ; (1 + A^)2

(2.35)

Consider the following function

(1 +

XZ)2

Then

(1 + X^)2

Thus Bup/(a:) = /(+oo) = 3. It follows that Sii(yo) < 4.

(2.38)

On the other hand, from (2.20) and (2.34), we have Fx2(2/o) = l + A + A(l + A 2 )^.

(2.39)

Together with (2.32) and (2.34) we get Ani{yo)

3A 3 (A 2 -1)[1 + A + A(1+A 2 )3] (1 + A2)f 3A (A - 1) - (1 + A 2 ) 2 3A4(A - 1) (1 + A)3 4

2

Put O = (0,---,0),

ei = ( l , 0 , - - - , 0 ) ,

( 2 4Q)

'

Geometric

Quantities

on a Minkowski

Space

e

4 l^mMI I^IUWI >> | 3^ A - ^(A-1) lffll(j/o)|5 ~ 8 (1 + A)

21

then e

A||o

e

IIA°,Wi>, ^i)l ,

>

e

e

IS(o,y0)( l> l)l*

= =

(2.41)

from (2.38) and (2.40) Proposition 2.5.3([Shen, 1998])

On i?" we de/me F : TRn -» [0, oo)

F : (z.y) ^ ^ > < | 2 + NI(£l2/T)*Then (Rn,F) cannot be isometrically into any embedded finite-dimen sional Minkowski manifold. Proof.

By (2.41) „ ...

m

^ 3 ||a:|| 4 (||a;|| - 1)

' * l

(i + NI)3 - + 0 °

... „

(W-+0°)-

Hence the Cartan tensor of (Rn,F) is not bounded. Now our conclusion is an immediate consequence of Corollary 2.5.2. •

This page is intentionally left blank

Chapter 3

Chern Connection

Besides the geometric invariants of Minkowski tangent spaces, there are several geometric quantities on a Finsler manifold. We will describe these invariants by introducing the connection on a Finsler manifold. From the point of view of differential geometry the most notable connection was Chern's connection defined in 1948, which is derived by using exterior differentiation on the Hilbert form [Bao and Shen, 1994; Chern, 1948], Chern connection is torsion-free and 'almost' compatible with the fundamental tensor. In this chapter, we will give the explicit construction of Chern connection and analysis the main properties of this connection.

3.1

The adapted frame on a Finsler bundle

Let (M, F) be a Finsler manifold and SM its projective sphere bundle. Consider the Finsler bundle p*TM of (M,F) (see Section 1.4). There is a global section / of p*TM. It is defined by p*TM

£

SM Fig. 3.1

23

24

An Introduction

1

to Finsler

Geometry

= hx,iy]) = ( ^ [y}> ~Eu—\)

We call Z the distinguished section. Let 5 := gijdx1 ® dxJ' denote the fundamental tensor of (M,F). Then 5 is positively homogeneous of degree zero with respect to y. Hence g G T(Q2p*T*M) where p*T*M is the dual Finsler bundle. Lemma 3.1.1 For any (x, [y]) G SM, there exists an open subset V and ex,--- ,en G Yv(p*T*M), such that (x, [y]) G V c -SM, g(e;,e^) = ^ and en = l\vProof. For each x G M, there exists a local coordinate chart (U; x1, • • • , xn) such that x G U. Thus ,^,lGTp-Hu)(p*TM).

dx^'"' Since I is nowhere zero and kx,[y])

z

y F(x,y)

yi d ~ F(z,y)cV'

(3.1)

we assume that yn > 0.

(3.2)

Thus there exists open subset V C p~l{U) of (x, [y]) such that yn\v > 0. It follows afi"! " ' i 9 x n-i ,1 are linear independent on whole V. Setting e n = l\v- By using the Schmidt orthogonalization, we get ex,--- ,en G Yy{p*TM) satisfying g(a,ej) — 6ij. For instance, P

-I

l

9(^,1)

d

dx"~

e n - i := £n-i/vS(£n-i>£n-i)We call the frame field in Lemma 3.1.1 is adapted. Lemma 3.1.2 / / w1, • • • ,un G Ty(p*T*M) is the dual frame field of a adapted frame, then ujn is the Hilbert form.

Chern

Proof.

Connection

25

From (1.12),(1.13),(1.15) and (3.1), we have Fyi{x,y)

=gij(x,y)F

y°

= gij(x,

y)dxJ(en).

Thus, for k — 1, • • • , n, we get u(ek) = [Fyi(x,y)dxi](ek) = gij(x,y)dxl <& dx](ek,en) - g(eic, e„) = Skn = ujn(ek). Notice that u{Xt[y]) G T*M, for any (a;, [y]) G V and j G {1, • • • , n } . The frame u>3 can be expressed in terms of dx^. That is uS:=v>k(x,[y])dxk.

(3.3)

Hence uj G IV(T*SM).

(3.4)

It is easy to verify the following lemma using (3.1),(3.3), Lemma 3.1.2 and gkl =Y,idxk{ei)dxl{ei). Lemma 3.1.3 Let {e,} denote the adapted frame field of (M,F) the functions defined in (3.3). Then

(0 -9x1 = ^ ^

and v\

(3-5)

vkn = Fyk,

(3.6)

(Hi)

^ W = < ^ ' ,

(3.7)

(iv)

vk5klVjl

(3.8)

(«)

(v) (ui)

vkayk

— gijt = 0,

^QVfc/3^=FJFy,yfc,

where l
(3.9) (3.10)

26

An Introduction

to Finsler

Geometry

Similarly, setting :=Ujl{x,[y\)—v

ej

and {u1} is the dual frame of {ej}, then we have Lemma 3.1.4 (t)

dxi = ufciwfc,

(3

(ii)

unl = yl/F,

(3

(Hi)

ukzgijUij = <5fcj,

(3

(iv)

UkWvti

=j*,

(3

= 0,

(3

(«)

(m)

Fykuak

ua3ui3kFFyjyk

=Sap.

(3

Corollary 3.1.5 i V u / = <$/; u / = 5jivkigkl;

3.2

ukivjk = S/,

(3

v^ = 6ijUjlglk.

(3

Construction of Chern connection

Lemma 3.2.1

Let UJ denote the Hilbert form of(M,F). du = 0

mod

u)1 A UJ3 ;

w a A cfa/^.

Then

Ckern

Connection

27

Proof. dw = d[Fyi (x, y)dxl] dFyi A dxl = (FyiXJdx3 + Fyiyidy>) A uklwk = Fy.^u^u^uj1 A wk + Fyiyjujdyi = FyiXJui:1UklLUl A uk + Fyiyiujdyi

(3.20) a

A u> + Fyiyj ^dyi A wa

Aw

from (1.13),(3.11) and (3.12). Lemma 3.2.2

•

There exist ujjn £ T(T*SM),

such that

dcj = UJJ A ujjn.

(3.21)

Moreover uijn can be taken as "n

uan = {ua3ui3kFykxj

(3.22)

— «,

u i + A Q/3 )a/ + -^r(Fxi

- ykFyJxk)co -

uakFyjykdyj, (3.23)

where Xa/s : V ( c 5 M ) —> i? satisfy that Aa/3 = A/3a. Proof.

By using (1.12) and (3.12), we have FyixjUv>UkllJ

Au)k

= F y i a . j ^ U a ' w A w ° | ^Uaj0Ja +UajU0iWa

= (FxiuJ/F)ua j

AW

A W'3) z

Awa

Fy,XJ £ V " * 5

A w

3

+FyixjUa up Lj

A a/ .

Substituting it into (3.20) yields (3.21)-(3.23).

D

Lemma 3.2.3 dua = w* A wi a

(3.24)

u ^ " = wfcadw/3fc + &%; + / / ^ . X

( 3 - 25 )

w/iere

Wn° = jvkadyk

+

tfw*

(3.26)

28

An Introduction

and yf'py :V-^>R

Proof.

to Finsler

Geometry

satisfying /xa/37 = ^ayi3, &a \V -* R.

From (3.3),(3.9),(3.11),(3.12) and (3.17), we have du>a = d{vkadxk) — dvka A dxk = Uikdvka a

A UJ1

= —Vk dui

k

(3.27)

A wi

= u0 A vkadupk + w A vkad(^r) = OJP A vkadupk + w A (vka/F)dyk. On the other hand, since /Ua/37 = /i a 7 / 3, we get a / A [fr a u; + lxalPuJ^] I w A &
(3.28) D

Taking £na = -6a°^-(Fx3

- ykFyix,),

(3.29)

&<• = - < 5 < * > ^ V * > ^ + Aff/3),

(3.30)

toan + 6apLjJ = 0,

(3.31)

then

t V = vkadupk

Proof.

- 5aiT(uJupkFykxJ

+ XrP)u> + iffrw"1.

(3.32)

By using (3.10) and (3.17), we have v a ua Fyjyk = -—.

(3.33)

Now it is easy to obtain (3.31) and (3.32) from (3.23),(3.25),(3.26),(3.29), (3.30) and (3.33). •

Chern

Connection

29

Lemma 3.2.5 = - ua3up%[d{FFyiyj)

upa +uap

+ (FyJxi + Fyixi)u)]

a

- 2Xpauj + {5a(Jfj, pry +

Sapfj,aa^)u'' (3.34)

Proof.

From (3.17) and (3.10), we have 5aaviadupi

+ SapViaduJ

= -u(rjUpid(FFyiyj).

(3.35)

On the other hand uPo + uap := u>paSaa- + uaaSap. Plugging (3.32) into (3.36) and using (3.35) yields (3.34).

(3.36) D

Lemma 3.2.6 Let V be an n-dimensional vector space, and assume that F : V —> [0, +oo) is positively homogeneous of degree one and (^-)yiyi is positive definite. Then (i)

F(y)>0

(ii)

for

FytyiC?>0,

y^O,

(3.37)

V£€V,

(3.38)

and " — 0" only if £• and y are collinear. Proof.

Setting

m ••= ( ? ) V

Z

,

(3-39)

/ yiyi

then gij:=FFyiyJ+FyiFyJ,

(3.40)

(1.12) and (1.13) imply that ga(y)yiyj=F2(y). Notice that gij is positive definite, we have (3.37). Also,

(3.41)

30

An Introduction

to Finsler

Geometry

is an inner product on V for any y e ^ \ { 0 } . By the Cauchy-Schwarz inequality

Ms>)£V] 2 < [gij(y)ee][9ki(y)vkv%

(3.42)

where equality holds iff £ and r\ are collinear. In particular, we take r\ = y and use (3.41) and (3.42) [9iJmiyj}2
(3.43)

where equality holds iff £ and y are collinear. Thus, from (3.40) and (3.43)

Fy^(y)ee = F-H^j -l Fy
3

F-1gij?&-F-1(F-1{gijyi&)2

= F- [F29iJee - to^')2] > 0 where equality holds iff £ and y are collinear.

L e m m a 3.2.7

•

Let M be a Finsler manifold. Put 6k = Fyjykdyj

e

T(T*SM),

then we have at least n — 1 linear independent forms among 6\, • • • , 9n.

Proof. Denote the complementary minor of Fyiyj in det(Fykyi) It is easy to see that

by fli.

A • • • A dj A • • • A 6n = Pdy1 A • • • A dyi A • • • A dyn.

(3.44)

Normalizing the homogeneous coordinate (j/ 1 , • • • , yn) we have y1dy1 + •••+ yndyn

= 0.

Consider a (x, [y]) e S'M such that yn ^ 0, we get

y

n

By using (3.44), (3.45) and (3.46), we get

(3.45)

Chern Connection

31

6>i A • • • A 6j A • • • A 6n

= f^dy1

A • • • A dyn~l

+faHyl A • • • A
•••

Fylyj-1

^ / " y 1 ' ' ' Fy^yi-1

T/ F y l ^ ' + l •••

Fylyl,

2/™ Fynyj+1

Fynyn

' ''

= n — 1. Since Fyiyjyl

Note by Lemma 3.2.6 we have rank(Fyiyj) L e m m a 1.4.1),

(3.47)

= 0 (cf.

2 / ( ^ 0 ) 1 ( F j , v , - - - ,Fj,nj,i)Hence there exist j £ {1, • • • , n } , such t h a t 6\ A • • • A 6j A • • • A 6n ^ 0. OJ1, u>an are the linear independent

Proposition 3.2.8 SM.

Proof.

Pfaff forms

on

Assume t h a t fiaLoan + ViU* = 0.

(3.48)

Substituting (3.23) into (3.48) yields »auak6k where 6k = Fyjykdyi

= 0,

(3.49)

(cf. Lemma 3.2.7). By (1.13) we have y181 + --- + yn-16n_l+ynQn

= 0.

(3.50)

From Lemma 3.2.7, we can assume t h a t 0i, • • • , # n - i are linear independent. If yn = 0 then (3.50) implies t h a t y = 0. This is a contradiction. Setting /j.auan

= Xyn.

(3.51)

Using (3.51) we can remove 9n from (3.49) and (3.50) a n d get

{^ui Since 6\,- • • ,9n-i

- \yP)0p = 0.

are linear independent, we get, together with (3.51) Hauak

= \yk.

(3.52)

•

32

An Introduction

to Finsler

Geometry

Thus 0 = nauakvkn

= \ykvkn

= XykFyk = XF(x,y)

from (3.6) and (3.52). Note that (x, [y]) 6 SM, Lemma 3.2.6 implies that A = 0. Substituting it into (3.52) yields /Ltauak

= 0.

On the other hand, since rank(u^) = n, we get rank(uak) = n—1. Without lose of generality, we assume that det(ua^) ^ 0. Thus iJ,auaP = 0 implies taht fxa = 0. Plugging it into (3.48) yields £ \ i>*a/ = 0. Hence Vi = 0. D Remark From Proposition 3.2.8, w1 A • • • A u>n A w\n A • • • A un-\n be considered as a volume form on SM. Setting I I : = SijLO1 UJJ + SapU)na

ca

(g> LJn13.

Then S M is a (2n-l)-dimensional Riemannian manifold with Sasaki-type Riemannian metric G[Bao and Shen, 1994]. 3.3

Properties of Chern connection

First, Chern connection is torsion-free from (3.21) and (3.24). Note sition 3.2.8, we put d(FFyiyi) = Kijau>an + GijkW , where K^a

and Gijk are symmetric with respect to i and j .

Proposition 3.3.1

In (3.25) and (3.30), we choose respectively

ApCT = - ^ u/u^idjn VP*a = -^^(ufj-uJGijv

+ FyJX, + FyiXJ),

- ujujGitf

+ uJu^Giji).

(3.54)

(3.55)

Then toPa + uap = -2Hp(jaujna,

(3.56)

Habc = AijkUalubjuck

(3.57)

where

n

Chern Connection

33

and (_d_ _d_ ~ \dx^dx^dx^

d

Aijk A

Proof.

Prom (3.53) -^T(FFyiyj)dyk

= Ki^w71

mod

dx\

(3.58)

Plugging (3.23) into (3.58) and comparing the coefficients of dyk yield — Ki^Ua Fykyl = FykFyiyj ~\~ FFyiyjyk.

(3.59)

Combine with (3.15) and (3.16), we have -Kija8aP

= F2Fyiyjykupk.

(3.60)

Substituting (3.15), (3.53)-(3.55) and (3.57) into (3.34) and using (3.60) yield Upa

+U>ap = -F2FyiyjykU()kU
= - H~pa aL0>na.

Remark (1) Habc is zero whenever any of its three indices has the value n. (2) (3.31) and (3.56) combine to give Wife + Wfc, = -HikjioJ.

(3.61)

It means that Chern connection is almost compatible with metric g. Lemma 3.3.2 Having spectifield Xpa in (3.54) > formula for u>a71, namely, wn" = -uasFy,ykdyk

i

?/

[ + uas[-—F 2~ yk(Grsn 1

-^(Gskn

2

we

have the following

+ Fyrxs + Fysxk

—

Fy.xr) Fykxs)]d:

k

(3.62)

An Introduction to Finsler Geometry

34

Proof.

Direct calculations give that ykFykxJ

= (yhFyk)XJ=FXJ

(3.63)

and k

ufvf

- unkVin =5ik-^Fyi.

= u, V

(3.64)

Plugging (3.54) into (3.23) and using (3.63) and (3.64) yields (3.62).

Lemma 3.3.3

Having specified Kija K?jUas

Proof.

= -ysFyiyJ

•

in (3.53), we obtain the following - F2FyiyJykgks.

(3.65)

From (3.18) and (3.60) K% = -vtaF2Fyiyjykgkl.

(3.66)

Together with (3.64) we have K^uas

= ysytgtiFyiyjykgkl

- F2Fyly]ykgks

= -ysFyiyJ

F2Fyiyjykgks.

-

• Corollary 3.3.4

We can rewrite the formula for K^uas KijUa

Proof.

-~9

*

jj-fck

as

•

dy

(3.67)

By using (3.6),(3.12) and (3.18) we have 9iJFj = | .

(3.68)

Together with (3.65) we get (3.67). Lemma 3.3.5 Giji =(FxkFyiyj

•

Having specified G^i in (3.53), we obtain the following + FFyiyjxk)ui

+ (ysFyiyj + F

\yk X \--p5nl{Gksn

Fyiyjyrgrs)

1 - Fysxk + Fykxs)

— -Ul

(Gskn ~ Fykx,

+

Fy,xk)].

(3.69)

Chern

Connection

35

Proof. Substitute (3.62) and (3.3) into (3.53) and compare the coefficients of dxk. Then we contract it with u/fc. Finally we plug (3.65) into this formula. •

Lemma 3.3.6

With the same notation, we have

Gabn = FFyayby.gc\Fx, Proof.

- ylFySxl)

+ -^ylFxl

+ ylFy«ybxi.

(3.70)

Using unk = £ and (3.69).

•

Lemma 3.3.7 g^Fyi(ykFyJxk-FXJ)

Proof.

= 0.

(3.71)

By using (3.63) and (3.68).

•

Put G = \F2, 2 '

(3.72)

G, = i ( y ' G y . x . - G I . ) ,

(3.73)

G* = / G , .

(3.74)

^

—

- . i

Definition 3.3.8 The local functions Gl defined in (3.74) are called the geodesic coefficient ([Shen, 2001]). Remark The projection of an integral curve of G := y%-^ - IG1-^ called the geodesic of (M, F).

is

Lemma 3.3.9 (*)

Gl = ±{y'Fx.Fyi+y'FFyix.-FFxi).

(3.75)

An Introduction

36

g dGf ^ -^l^Lst=-±'^l± k FJ? dy n„.k

(„) \ J

to Finsler

Geometry

ldGs gtl ^(y-Fvlxa-Fxl) k + x Fc1 dy Ftmk 2o xa yx ' X (FykFySyt

+ FFySytyk

+

FysFytyk).

(3.76)

Proof, (i) A direct calculation. (ii) By using (3.68), (3.71) and (3.75), we have Gl

=

TW

Y{yaFy,xa~Fxl)x{FykFy°vt+FFyaytyk+Fy°Fytyk)-

(3 77)

-

Now (3.76) is an immedinat consequence of (3.74) and (3.77). Lemma 3.3.10 1 yr 7 : ^ F

y

k ( G

n

•

1 + Fyrxs

r s n

BC*

+

Fysxr-)

— - (Gsfc„ +

Fysxk

+

Fykxa)

1F

= - y ^ + --f{FFykytg«(Fxl

- y'F^)

+ Fxk + y*Fv>xr} (3.78)

Proof.

It is easy to see y Fyiyj = 0,

ylFyiyjxk

= 0;

y Fyrxs = Fxs.

Together with (3.70), we obtain (3.78). Theorem 3.3.11 re-expressed as wn

•

(i) The one forms u)an on SM can be more compactly

= -un

Twd*+F*'vhdyk

(3.79)

(ii) Let v : T[p*TM) -» T(p*T*M p*TM) denote covariant differentiation, onp*TM, relative to the Chern connection (w^), i.e. Vefc = Wfc* ® ej, then y is metric compatible if and only if (M, F) is Riemannian, hence y reduces to usual Levi-Civita connection.

Chern

37

Connection

Proof, (i) Plugging (3.78) into (3.62) and using (3.15) we obtain (3.79). (ii) y preserves inner product if and only if uty = Uji = 0. By using (3.61), it equivalents to Hijk = 0 . D 3.4

Horizontal and vertical subbundles of SM

Denote the fibres of SM by SXM := p _ 1 (z)- Thus TSXM = {J = 0}, where w1, • • • , uin is the dual frame field of a adapted frame. Hence H := {b £ TSM\ujna(b)

= 0}

is orthogonal to TSXM with respect to Riemannian metric II. We call H the horizontal subbundle of SM. Write \J TSXM by V, we call it the vertical subbundle of SM[Mo, 1998; Mo, 2000].

This page is intentionally left blank

Chapter 4

Covariant Differentiation and Second Class of Geometric Invariants In this chapter we will introduce the second class of geometric invariants on a Finsler manifold. These quantities describes t h e changing rates of t h e first class of geometric invariants along geodesies, or Hilbert form. First, we define the following.

4.1

H o r i z o n t a l a n d v e r t i c a l covariant d e r i v a t i v e s

Let (M,F) be a Finsler manifold and w1, • • • ,u)n the a d a p t e d frame fields on the dual Finsler bundle (cf. Lemma 3.1.2). Let Wi3 be Chern connection 1-form. T h u s w1 A - - - A w n A w n 1 A - - - A w / " 1 ^ 0 ,

dioj =uk

SikUjk

Awkj,

+ 6jkL0ik = -2Hijau>na.

(4.1)

(4.2)

(4.3)

By using u>^, we can define the covariant differentiation of t h e tensors on P*TM. For instance, let / : SM -> R be a smooth function. Write

Df := df := fw* + f.acna,

(4.4)

we call \i (resp. ; a) the horizontal (resp.vertical) covariant derivative. Recall t h a t H = {wna

= 0}, 39

V = {u/ = 0 } .

40

An Introduction to Finsler Geometry

Consider the Cartan tensor Hija on (M, F) and define DHija '•— dHij dHija — 2-i 22,Hkja^>i ija — Hkja^i — 2^ Hika^j a ~

k—

2^ Hijk^a

(4.5)

Usually, we call \n the covariant derivative along the Hilbert form. Similarly, we can define the covariant derivative for other type tensors on p*TM.

4.2

The covariant derivative along geodesic

Let II be the Sasaki-type Riemannian metric on SM and b : T(SM) T*(SM) the musical isomorphism induced by G, i.e.

Ii{X,Y) = X\Y).

(4.6)

Denote the horizontal left of y by y. Then

J

(4.7)

=0,

In fact y =y

d dxi

dGj d dyi dyi

(4.8)

2dgu k

_ dffjfc dxl y y

(4.9)

and Ji G3 = -g 4y

dx

Note that G3 are the geodesic coefficients [Shen, 2001], i.e., we have the following Proposition 4.2.1 l

\g \y)

F2 {^)yixk{y)y d

= \
0

9ji,

F2 -(-g-My) s

dgjk

yjyk-

(4.10)

Covariant

Differentiation

41

Proof. *- dxk

y

dxk

^

d dxk F2

=£

— )

V yjyl

F2 z

ylyi

'

IW-lrCwOXVl-B'' dx dx l y y

l

These formulas imply (4.10).

D

A C°°-curve c(t), t € / , is called a geodesic if the canonical lift c(t) in TM is an integral curve of the vector field a vS^-2C i 9rr dyi

L e m m a 4.2.2 smooth function.

(4-11)

Le£ c(t) 6e a geodesic on (M,F), Then, on (x,y), we have Fdf = — [f{c{i))]t=ou

mod

ua,

and f : SM

un

where c(0) = y. Proof.

It is sufficient to show that, from (4.7)

y(f) = Jt[f(c{t))}t=o. In fact,

df

W

of dyiK)dyi}

R a

An Introduction

42

to Finsler

Geometry

from (4.8),(4.11) and the following

Remark Sometimes we denote the covariant derivative along any geodesic, i.e. the Hilbert form, by " • ". For example, we have / := f\n and Hija := Hija\n, ect. Note that / : SM —» R is a function and Hija is a three order tensor.

4.3

Landsberg curvature

Let Hiia be the Cartan tensor of {M,F)(ci. La/3j

'•=

(3.57)) . Put

—Hap7

and L := ^^ Laf3 : M S-> (N, h) an isometric immersion. Let V denote the Levi-Civita connection for h and V the LeviCivita connection for induced Riemannian metric .Y = MVxY) ioi X,Y

eT(TM),

+

B(X,Y)

where

^(Vx^e^JM)

and B{X,Y) G r ( T x M ) .

We call B the second fundamental form for 0 and trace^-hB the meon cwvature for 0, denote by iJ. (/> is called to be minimal (resp. totally geodesic), if its mean curvature (resp. its second fundamental form) vanishes. It is easy to see that : M S-> (N, h) is totally geodesic if and only if <j> carries any geodesic on M into the geodesic on N. The Landsberg spaces have the following interesting geometric characterization.

Covariant

Differentiation

43

Theoem 4.3.2([Ai, 1993]) Let (M,F) be a Finsler manifold and SM its projective sphere bundle. Then (M, F) is of Landsberg type if and only if all projective spheres in SM are totally geodesic. Similarly, set •Ja •=z

Va\n

=

Vet

where r\ = ^,ar]aea denote the Cartan form. We call Jawa Landsberg curvature and denote it by J. Definition 4.3.3 if J = 0.

the mean

A Finsler manifold is said to be of weak Landsberg type

Z.Shen gives following geometric characterization of weak Landsberg manifold T h e o e m 4.3.4([Shen, 1994]) A Finsler manifold is of weak Landsberg type if and only if all projective spheres in the projective sphere bundle are minimal. Let Vol(x) denote the volume of the unit Finsler sphere ([Bao and Chern, 1996])

{y£TxM\F(x,y)

= l}.

For functions a1 — ai{x), we have

where n

I

dV := y/det{gij)

^

{-\)k-1ykdy1

A • • • A dyk A • • • A dyn.

fc=i

It follows that Theoem 4.3.5([Bao and Chern, 1996]) manifold. Then Vol(x)=constant.

Let (M,F)

be a weak Landsberg

An Introduction

44

Example

to Finsler

Geometry

Define F : TR2 -> [0, oo) by := ^p* + q2 + \(x)(p4 + q*)

F(x,y)

where y = (p, q) € TXR2 and A is a smoothly non-negative function. It's easy to see A (-„..-> -

I

Wv)-\ \ (cf. Example 1.3.2). Hence

, Ap 2 (p 4 +3q 4 )

-2ApV

(P 4 +9 4 )*

_2APV (p4+q4)I

y/det{gij) = J:1 + A

(p 4 +? 4 )5 1 |

\ _ 1

A,2(3p4+g4) I (p4+q4)t /

^JV)1 + 4 4 (p +
A 2 .4W 4 P +Q

By using (4.12), one finds that as the value of X(x) is increased from 0 to oo, the value of Vol(x) decreases monotonically from the expected 2rc to 5.4414 ([Bao and Shen, 2002]). D.Bao and S.S.Chern give the Gauss-Bonnet theorem for Finsler spaces whose Vol(x) are constant. Together with Theorem 4.3.5 we have the following T h e o e m 4.3.6 Let (M,F) Landsberg manifolds. Then

be an oriented compact 2k-dimentional weak

where [X] : M —* SM is an arbitrary section with isolated singularities and Pf(Q) is the Pfaffian of the curvature for the Chern connection and T the additional term in [Bao and Chern, 1996].

4.4

S-curvature

Let (M, F) be a Finsler manifold and T its distortion. Put S = fF, we call S the S-curvature of (M, F). Clearly S(x, \y) = \S{x,y),

A > 0.

(4.13)

Covariant

Differentiation

In a standard local coordinate systerm (x',y''), the geodesic coefficients of F. Lenmma 4.4.1

The S-curvature of(M,F)

let Gi — Gl(x,y)

2 ' t*y

denote

can be expressed as

k lq^vk-2F-'AkG -4,^. a(x) dxi

S== 9 J

Proof.

45

(4.14)

Prom Lemma 4.2.2 S(C) =

Tt[T{c(t))],

where c(t) is a geodesic of (M, F). By the definition of distortion, we have 5(c)

det d_ In \/ (9ij(c)) a(c) dt

d_ In^Jdet(gij(c)) dt

\ - — (lna(c)).

A similar calculation of (2.2) gives

*<*> = L/' Wgf ( ^ + ?('*-'*»*

- |y^(c).

By geodesic equation & + 2G\c) = 0, hence

where Afc is the Cartan form.

•

Definition 4.4.2 Let (M, F) be a Finsler manifold. F is said to have isotropic S-curvature if 5 = (n + l)cF, where c = c(x) is a scalar function on M. In particular, it is said to have constant S-curvature if S — (n + l)cF for some constant c.

An Introduction

46

Example 4.4.3

to Finsler

Geometry

On the unit ball Bn the following Randers metrics

F tr „\ - y/\y\2 - (\x?\y\2 - &y)2) + (*.y> + K

^

y )

~

±

i_|x|2

l-\x\*

±

(a>y)

l + (a,x)'ye

„^ T ™ x

have constant S'-curvature | ( n - f 1), where a £ Rn is a constant vector with \a\ < 1. Take a = 0. The resulting Randers metrics are the well-known Funk metrics on the unit ball 5™ [Mo, 2005]. Example 4.4.4

Let ( be an arbitrary constant and (iT,

if C > 0 ;

n = I } rr\

~

(4-15)

| B ( y 4 ) , if C<0. Define F : TCI -> [0, oo) by a ( i | j ) : =

« H ^

(4l6)

and

«*•»>=-r2?&-

(4 17)

-

where e is an arbitrary positive constant, and K is an arbitrary constant. Thus F has isotropic S'-curvature (cf. Proposition 4.4.6), i.e. S = (n+l)c.F where 2[e+(eC + K 2 )|z| 2 ]'

(4.18)

Notice that when C = — 1, K = ± 1 and e = 1 our metrics reduce the famous Funk metrics. The following result gives an equivalent condition for a Randers metric to be of isotropic S'-curvature. Proposition 4.4.5([Chen and Shen, 2003]) Let F = a + (i be a Randers metric on an n-dimentional manifold M where a = y/aij(x)ylyi and (3 = bi(x)yl. Then F has isotropic S-curvature, i.e. S = {n + l)cF if and only if nj = 2c(x)(a.ij - bibj) - biSj - bjSu

(4.19)

Covariant

47

Differentiation

where r

ij

+ bi\i)'

•= ^ I j

Si

s

ii

:= bkakjSji,

(akj)

9b

L

'•= 2 ^

i

b

^i)

(a^)'1

=

L

~

k

and •yfj is the Christoffel symbol with respect to the Riemannian metric In particular, if (5 is closed, then S = (n+ l)c(x)F holds if and only if bAj = 2c{x){aij

- bibj).

a.

(4.20)

Proof. For the sake of simplicity, we choose a n orthonormal basis for TXM such t h a t a,j = 6ij. Let p := lny/1

- W\l

(4.21)

and dp = Pidx1, i.e. Pi =

EbJbj\i

i-Ml'

where \\(3\W is defined in Section 1.3. According to (20) in [Shen, 2003], we have the following formula for 5-curvature, S = {n + l)(P - dp)

(4.22)

where P

-2asiyi).

•= T^injyV

It follows t h a t S — (n + l)cF if and only if rijyiyj

- 2aslyi

= 2(a + P)piyi

+ 2c(a + f3)2.

(4.23)

(4.23) is equivalent to the following two equations rij = bjPi + bipj + 2c(aij - bibj), -Si =

Pi

+ 2cbi.

(4.24) (4.25)

An Introduction

48

to Finsler

Geometry

First we assume that S — (n+l)cF. Then (4.24) and (4.25) hold. Plugging (4.25) into (4.24) gives (4.19). Now we assume that (4.19) holds. Note that

E sihi

=

E

3

biSi h

i J = °-

i,3

Contracting (4.19) with bj yield

X V ^ = M1-Il/3||^-Il/3||^

(4.26)

that is,

We obtain

It follows from (4.22) and (4.27)

Si =

-2c6, + i (J2 bAn + 13]}f| E &^l<) = " 2 c 6 < - *• ( 4 2 8 )

Thus ryi/V' - S a s i ^ = 2 i W + 2cF 2 ,

(4.29)

we obtain S = (n + l)(cF2 + ptf - prf) = (n + l)cF.

Proposition 4.4.6 Let F = a + (1 : Til —> [0, oo) be any function given in (4-16) and (4-17). Then F has the following properties: (i) F is a Randers metric and (3 an exact form; (ii) F has isotropic S-curvature, i.e. S = (n + l)cF, where c is given in (4.18). Proof.

Set u>:=l + (\x\2,

2 e

: = e C + K2

a2 = aij2/V, P = biy\

(4.30)

(4.31)

Covariant

Differentiation

49

Then e5a

Hi = - 2 - +

K2X1X:>

,

J—, fc =

Kxl

.

4.32

Since 1 + CM > ^> it is easy to see that if y ^ 0 then , _ K2(x,y)2

+ e\y\2 + e(\x\2\y\2

e(l + C|x| 2 )M 2

Moreover a(x, y) = 0 if and only if y = 0. It implies that a is a Riemannian metric. Write (a«) = (ay)" 1 -

(4-33)

By using (4.32), we have + -^-)afcj = —

*? = a*a*> = £ ( _

+ — ,

(4.34)

where ^ ^ i V

5

' .

(4.35)

i

(4.34) is equivalent to eujaij =L026Ji-K2xitj.

(4.36)

From (4.35) and (4.36) we have ecotj = ew Y2 xiaij

= u2xj - K2\x\Hj.

(4.37)

i

It implies that

t3

u>2xj

2cw2xj

(4 38)

= 7T7W = " ^ - '

-

where c :— c(x) is defined in (4.18). Substituting (4.38) into (4.36), we get a« = —{wHi l - K2xHj) = -(6j8 - K2X1^ I0) ew e e + g2\x\2

= -(5ij e

- 2cKxixj).

(4.39)

Using (4.32) and (4.39), we have m

i

= a-bibj

= ^

- 2 c ^ V ' ) ^ ^ = ^ J ^

< 1

by 1 + CM 2 > 0. It follows that F := a + (3 is a Randers metric.

(4.40)

50

An Introduction

to Finsler

Geometry

Consider f3 as a 1-form, then

?=«?*£*?„2

=d

1 + C M2

^ n ( l + CaWa)

T h u s (3 is exact. In particular, we have d/3 = 0. From (4.30), we get dx%

=

(4.41)

2(x\

Together with (4.32) we have

dan

k•¥•

dx

K:

4Cfts

2eC,

= —2 (zl<5jfc + xj6ik)

^SijX

w3

xlxjxk.

(4.42)

This implies t h a t _ Hik--

1(daij

2[dxk

daik +

_

dajk, <

dxi

dxj

= —SjkX1 - -^(6ijxk w " a;

2C« 2

6ikxj)

+

(4.43) x%xPxk.

By using (4.39) and (4.43), we get Ijk

=

a

Q25jk Tijk

=

e+

(4.44)

g2\x\2

Note t h a t 6, satisfies (4.32), we have 2 u i22 r Q„ 2 \x\ Sij

2(xlx^

€ + Q2\x\2

(4.45)

U!

and

3^ dxi

u)5ij — 2£xlxj =

(4.46)

K-

w

Here we have used (4.41). By the definition of 6JU, we get enSn k l i

2

2ecr 2

=

~ u(e + Q \x\ ) ~

u

On the other hand, from (4.32), we have

This implies t h a t bqj = 2c(x)(a,ij -

bibj),

On. Uij

(4.47)

Covariant Differentiation

51

where c(x) is defined in (4.18). Note that f3 is closed. From (4.20) we get F has isotropic S-curvature. •

This page is intentionally left blank

Chapter 5

Riemann Invariants and Variations of Arc Length

Besides the non-Riemannian quantities, there are several important geometric quantities in a Finsler manifold. These invariants are the natural extension of geometric quantities in a Riemannian manifold. We call them the Riemannian invariants. The flag curvature is the most important Riemannian quantity in Finsler geometry because it lies in second variation formula of arc length and takes the place of the sectional curvature in the Riemannian case. In this chapter we will introduce Riemannian invariants in a Finsler manifold by discussing the Chern curvature 2-form. Then we will give the first and second variation formulas of arc length.

5.1

C u r v a t u r e s of C h e r n connection

Differential (4.2) and using (4.2) one deduces 0= = = =

d2cji d(ujkAukj) (duik) A uki -ujk A dukj — u>l A [dwj — u>ik A ujkj}.

. {

. '

;

Put JV := dcjij - uJik A u;kj G T(A 2 5M).

(5.2)

We call fljJ' the curvature 2-form of Chern connection. By (4.1), ftjJ can be expressed by *V := ^iV'ww* A J + Pijkauk

A wna + Qijal3LJna A tone. 53

(5.3)

54

An Introduction

to Finsler

Geometry

Prom (5.1), (5.2) and (5.3) we have 0 = - w i A £lij = — wl A Qijapuna

Aw /

mod UJ1 A cuh A LO1 ,

w'AwfcAw„a.

It follows t h a t Qjcxpuj'1 A uina A uj* = 0. T h u s we obtain Qi\f3

= Qi-'pa, i-e-

Substituting this into (5.3) yields J V = ^/V'fcjw* Aw' + Pi j f c a w f c A w „ a

(5.4)

we agree on Rijki

= -Rijik-

(5.5)

D e f i n i t i o n 5.1.1 ([Chern, 1992]) Let (M,F) be a Finsler manifold. We call RiJki (resp. Pi°ka) the Riemannian (resp. Minkowskian) curvature. Plugging (5.4) into (5.1) yields - y ^ / V ' f c / o / A w ' Aul

= 0

mod w' Awfc A w n a .

T h u s one gets the First Bianchi Identity Rijki + Rkju

+ Rijik

= 0.

(5.6)

Similarly, from (5.1),(5.2) and (5.4), we have ^2 Pijkaul

A ujk A u>na = 0

modw'Aw'Aw'.

Hence the Minkowskian curvature satisfies t h a t * i ka

=

*k ia-

\p-' )

Setting u>ij = ^2SjkU>ik. Substituting this into (4.3) yields oJij + uiji = —2 2_^ Hijauina,

(5.8)

Riemann Invariants

55

where {Hija} is the Cartan tensor of (M, F). Differentiating (5.8), we have dun + dun = - 2 ^2(dHija)

A w£ - 2 5 3 Hijadw%.

(5.9)

a

Put ilij = ^25jk£lik. From (5.2), we obtain du>ij = Clij + 5 3 uik A wfcj,

dw„a = ftna + 5 3 Wn" A W/3a (•.• w„" = 0). Plugging them into (5.9) yields SUj + flji + 2Hijknnk

+ (/) = 0,

(5.10)

where

(-0 = 53Wifc A Wfcj + 5 3 ^

A Wki +2

y^AdHv<x)

A w Q

™

en)

+ 2 53ijy a w„ /3 Aw /3 a . By using (5.8) we get (II) = Y^k Wik A (~Ujk - 2 2 Hkja^n") + Hk Wik A (-Wife - 2 X) HkiocOJn") = - 2 J2k(HkjaUik + HkiaWjk) A W„ a .

Thus (/) = 2 53(£>tf ija )A W „a, a

where {DHija) is the covariant derivative of Cartan tensor defined in (4.5). Plugging this into (5.10) yields

fiy + VLji + 2 5 3 JJykOi* + 2 53(Dff i i a ) A w n a = 0.

(5.11)

56

An Introduction to Finsler Geometry

Together with (5.4) and (4.5) we have 0 = ~RijklU)k Aw1 + PijkaoJk

A L0na + -RjikioJk

A u ' l Pjikauk

A w„

'1 ( ^ " w W * A W ' + Pnak(3UJk A 0 , /

+ 2Hija

+ 2 (Hija^U)

+ HijaikOJn

) A WnC (5.12)

It follows t h a t (Rijki + Rjiki + 2HijaRnaki)tok

mod wfc A w n Q , wna A w / .

AJ = 0

By using (5.5), we have Rijki + Rjiki + 2HijaRnaki

= 0.

(5.13)

We can re-express (5.12) as (Pijkoc + Pjik* + 2Hijf3Pnla

+ 2HijalkJ

cok A a £ = 0

mod u / A coj,

Hence -Pijfca + Pjika

+ 2EHijpPn

ka + 2^?jj a |fc = 0.

(5-14)

Finally, we take the components of u)na A uinP in (5.12) and obtain

T h e following Proposition tells us the Minkowski curvature is a nonRiemannian quantity: P r o p o s i t i o n 5.1.2 follows Pijka

Proof.

=

~ti%jpL

The Minkowski

ka T tlkifiLi

curvature

ja ~ tlikpl-/

ia — tiija\k

Putting T-n *ijka t •CJijka '•=

Pijka

~r + jika ~

can be represent

i *ikia\j

~

as

"-jka\i'

Riemann

Invariants

57

and using (5.7) we have p

*ijka

r ljika

ijka -

*jkia

g i-'ijka

t *kjia

+

*kija

T"

2

i *-Jjkia

likja

2

(5.15)

^kijot-

From (5.14) we have ^ijkot

~

tlij/3*n

ka

^-ija\k'

Plugging it into (5.15) yields *ijka

~

tlij{3*n ka **ija\k ,TT p /3 , TT ~Tn-kif3in ja ~r "fcialj •

^jkf3-^n

ia ~~ *ijka\i

/r -t r-\ (O.IOJ

On the other hand, from (4.5) we have HnialjU3

= dHnia

— Hkia^n

~ Hnka^i

— Hnik^a 13

= -HpiaLOn

m o d U)n = 0

m o d OjJ3

It follows that HniaU = 0.

(5.17)

Combining with (5.16) one yields that •< n/3na —

*lnf}^rn

Jlnf3a\n

na

tlpny-^n

na

"/3raa|n

= 0

(5.18)

Together with (5.16) and (5.17) we obtain *njka

~

-^nj/3-^ri ka <-Hkn/3*n

=

H-jka

=

ja

^nja\k '

tljkp*n

na

fljka\n

*lkna\j

^jka'

Substituting it into (5.16), we have Proposition 5.1.2.

•

Definition 5.1.3 Let (M,F) be a Finsler manifold. If its Minko wskian curvature vanishes, we call (M, F) & Berwald manifold.

An Introduction

58

to Finsler

Geometry

It is easy to see that all Berwald manifolds are of Landsberg type. Z.Szabo showed the following: Theorem 5.1.4 ([Szabo, 1981]) Any Berwald surface is either Riemannian or locally Minkowskian. The following question is still open: is there any Landsberg manifold which is not of Berwald type? 5.2

Flag curvature

Let (M, F) be a Finsler manifold, and RJ u its Riemannian curvature. The flag curvature tensor is a two order tensor defined by •KQ/3 : = oajiin

pn.

It is easy to see Rap is symmetric from (5.5) and the First Bianchi Identity. Definition 5.2.1 by

The eigenvalues of the flag curvature tensor, denoted

«1 < • • • < l^n-l

are the most important intrinsic invariants of the Finsler metric. We call Ka the a.-th principal curvature. Definition 5.2.2 The trace of the flag curvature tensor, i.e. called Ricci scalar. Usually we denote Ricci scalar by Ric. Ricci scalar is a function on SM. Definition 5.2.3 Let y £ TxM\{0} and P a 2-dimensional linear subspace of TXM which includes y. Taking b € P such that 9(x,[y])(b, y) = 0, a

g(x,[y]){b, b) = 1.

a

Then b = b ea. The quantity Rapb b^ is a function of the flag {x,y,P}, denoted by n(P,y) which is called the flag curvature at {x, [y],P}. Definition 5.2.4 We say that a Finsler manifold (M,F) is of scalar curvature if there is a scalar function K on SM such that for any (x, [y]) G

SM the principal curvatures na = n, a = I,- • • ,n-l.

In particular, we say

Riemann

Invariants

59

(M, F) is of constant (flag) curvature K if it is of constant scalar curvature K.

It is easy to show the following: Proposition 5.2.5 If(M, F) has scalar curvature K = K(X) and dim M > 3, then (M, F) is of constant (flag) curvature.

5.3

The first variation of arc length

In this section, we use the method of differential forms to describe the first variation. Consider the rectangle

A:=[to,*i] x h M l Let (M, F) be a Finsler manifold. We map it into M using a : A —> M, (t,u) H-> a(t,u) such that the i-curves (u = constant) are smooth. One obtains two vector fields defined over the square: T:

=a*'!]'

U:=tr

-\eu

(*o,l)

(5.19)

(*i,l)

A *-t

(to,-l)

(*i,-l)

Fig. 5.1

T gives the velocity vectors to the ^-curves. Strictly speaking, T and U are maps from A into TM; but we will occasionally find it convenient to regard them, by a slight abuse of notation, as maps from A into p*TM. The map a admits a lift a : A —> SM, defined by a(t,u):=(a(t,u),T(t,u)).

(5.20)

An Introduction

60

to Finsler

Geometry

Fig. 5.2

SM v A — - M Correspondingly, one gets the following vector fields over

CT(A):

(5.21) For any (t, u) € A T(t, u) « (or(t, u), T(t,«)) G p T M . Put ||T(^U)||:=v^(^T)

(5.22)

nt,u) = T -

Then, from (5.22), we have \\T(t,u)\\ = JTWgij

=

F(a\.-.,an-T\---,Tn).

Hence T = \\T\\en =

F{a,T)eri

(5.23)

where {e,} is the adapted frame field. With respect to {ej}, we obtain the dual co-frame field {UJ1} and the connection 1-forms {uV}. These 1-forms

Riemann

Invariants

61

can be pulled back to A by cr*, i.e. a* : T*SM —• T* A . We write a*(wi) = aidt + bidu,

(5.24)

d*{u>l) = 4dt + Vidu. Denote a by (cr1, • • • ,an).

(5.25)

Thus T •

dax d dt dxl

Prom (5.20) dan.

„ da1

, i

-

Togother with (5.21) we get

T=^!A

+

^ A =T at 2 dyJ

at c V

mod

A

°

cV

m

Since wi € T(p*T*M), one has

ui{t)=w\T). Similarly, we have ui{U)=ui(U). Together with (5.21) and (5.24), we have a* =

cfdtijt)

= ^(f)=u/(T) = |mk(e„) and

Hence we get aQ=0,

a n = ||T||;

bi = Ui.

(5.26)

62

An Introduction

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Geometry

Since wnn = 0, (5.21) and (5.24), aij=uji;>{f),

ann = 0,

V=w/(C7),

bnn = 0

(5.27)

and, from (3.31)

aan = -Sa/3aJ,

ban = -8aPbJ.

(5.28)

Using cr* to pull back the torsion-free condition (4.2), we have d(crV) = a*wj A cr*w/. From (5.24) and (5.25) d(cr'V') = diatdt + Wdu) = (—^- + -^-)dt A du du at and 5*u)j A <J*(cV) = (ajbjl - Vaj^dt

A rfu.

Thus one gets Bai

Bti

Together with (5.26) and (5.27) dba _

~dT~ dan

anbna-bjaja,

(5.30)

Bbn

From (5.2) and (5.25) cr'(JV) = ^ ( d w ^ - ^ ' A w / ) = ( - ^

+ ^

- a f c J V + bk'a^dt

Adu.

On the other hand, by using (5.4),(5.5),(5.24),(5.25) and (5.26), ff*(fV) = d*{\Rk'it^ AUJ1 + Pj^iJ A Wn/J) j = (|i?fc^a 6' iRk'jiaV +PkijpaibnP - Pk^pVa^dt A du n l = {Rk\ia b + PJnpaTbJ - P^jpVa^dt

A du.

Thus we obtain

^f ~ ^T

= ak3bjl

~ hJaji

+ i?fci

"' a " 6 ' + Pk\panbJ -

Pk^VaS, (5.32)

Riemann

Invariants

63

in particular, specializing k to a, i to n, and using (5.18) one finds that Pann/3 = 0 and 8a " 8b n -£= -^- -

a

« ' V + be?«in - Ranmanbl

+ PcTipVaJ.

(5.33)

Let
= ft1 \W*TTt\\dt

= 2 mi* =

/

,

>

"

*

,

(5 34),

'

•

The formula for the first variation of arc length now follows from (5.31) and (5.34)

£'(«) = £ J i o a n d * = It' TT 1 ^ = ftxM +

bv)dt

(5 35)

"

Suppose that all the t-curves have same end-points, i.e. Imato = x,

Imati = y.

(5.36)

Then

from (5.19). Similarly U(ti,u) = 0. Thus we get , from (5.26) bn\%=Un\ttl=u,»{U)\%=u>n(U\%)=0. Substituting into (5.36) yields L'(u) = / * baaandt. 'to

(5.37)

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An Introduction

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Geometry

L e m m a 5.3.1 Let (M,F) be a Finsler manifold and c : [to,ti] -+ M a curve on M. Then c is a geodesic if and only if for any a : A —• M satisfying (5.36) and a(t,0) = c(t), we have L'(0) = 0. Proposition 5.3.2 Let r : [£o,*i] be a curve on a Finsler manifold. r is a geodesic on M if and only if ana = 0 where f*uina

Then

= anadt

{u>ia} is the Chern connection 1-form and f satisfies r = p o f. Proof.

Taking a variation of r with fixed end-points p[expT(t)(^2e(t)aanuea)],

<j{t,u) =

where e : [£o,*i] —> -R is differentiable and e |(t 0l ti)> 0> e(^o) = e(*i) = 0. Thus a(t,0)=T(t),

U = a*^- = Y,
From (5.26)we get rti

0 = L'(0)= / ' B O "2e(t)dt. Jt0 n

It follows that aa (5.37). 5.4

a

= 0. Conversly, if aan = 0, then r is a geodesic form of D

The second variation of arc length

The setup is just like we had in Section 5.3, except now the base curve o~{t) = cr(t,0) in our variation cr(t,u), with (t,u) G A := [to,*i] x [—1,1]. From (5.33) d -(baaan) du r dba,a n^,adaa b uaa + o du du n dba n a Bb ^-aa + b -£- - ajbjn du

= ^aban)

+ bjaf

+ aan[^- - PfftfV

- Rannlanbl

+

PanjfsVan?

+ b%a\ - anbablRannl + (I),

Riemann

Invariants

65

where (/) := - ^ b

n a

- baajbjn

= bnanaban

-

anbnaba

from (5.27) and (5.30). Therefore d'an _ _d_[dbZ j_uan n] " 5 ^ ~ du 1 dt "^ " " « J d_db a n dt 8£ + &(& &a ) n a

n

n

-b S % ]

+ « a n [ ^ - PfTfhPV a

n

- a (bn ba

a l

+

+ bPbp"

(5.38)

n

b b Ra ni).

Note that a(t) = cr(t,0) is geodesic, ana = 0 from Proposition 5.3.2. Thus we obtain the formula for the second variation of arc length along a geodesic L"(0) =

£L(u) 9

4fdt

=£

(5-39)

For the remainder of this section, we shall discuss the covariant differentiation along curves and re-express variation formulas mentioned above. Let 7(s) be a curve on M with velocity S(s); let W(s) be a vector field in M defined along 7. Corresponding to any lift 7(5) := (7(s), y(s)) of 7 from M into 5 M , we set S := 7 , ^ = •^("f(s),y(s)) and define the following Denote Chern connection 1-form by {wt 1 }, and UJ

Definition 5.4.1 by wfcefc. We call

DSW :--

dWl

ds

+

wW(S)

the covariant derivative of W along 7 (with lift 7). From (4.3), it is easy to show

(Dsg)(V,W) =

VWHupui^iS),

where Dsg is defined by (Dsg)(V, W) = ^[g(V, W)\ - g(DsV, W) -

g(V,DSW)

for W,W € T(P*TM). We now mention two special cases in which this term drops out, i.e. (Dsg)(V, W) = 0:

An Introduction

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to Finsler

Geometry

1. if V or W is proportional to y; 2. i f ( u ; n % ( S ) = 0 . In this sense, one can say Chern connection has just the right amount of metric-compatibility. Return now to the rectangle a(tu), a : A —* M. We lift it into SM using the velocity T of the t-curves, namely through a := (a, T). One finds that DTU = DVT,

(5.40)

where U = cr*(^). By the definition of the curvature form, we have

£V(t/,f) = | W ( f ) - -^(U)

-ukj(UW(f)

WCZ>/(£).

Hence DVDTZ

= DTDuZ

+ ZkSlk\U,

f)ei

(5.41)

for any vector field Z(t, u) in c(A). On the other hand

nfc«(tf,f) = f^fcV^' A J + p fc W' AUnaJ (u,f). Thus we obtain DC/DTZ

= DTDuZ

+ Z^Rk'jnU^TW

+

PkiJaUjwna(f)

-Pfci„a||T||a;„Q(^)]ei

(5.42) By using (5.40) and metric-compatibility we have the first variational formulas in terms of covariant derivatives

= i::{m[9(U,^-l)}-9(U,DT(^))}dt.

[

A6)

The equation for a geodesic is thus DT(^-)=0.

(5.44)

If the geodesic is given a constant speed parametrization, then (5.44) reduces to the following form DTT = 0.

Riemann

Invariants

67

A direct calculation gives, from (5.43) t=ti

i"(o) -

»

|JW

rt

M)

+ g (if, [DTDu - DVDT} (~\

\\dt

(5.45)

By using (5.23) and (5.42)

PnijaU*wna(f)

= [-i^^^HTII -

+

Pnina\\T\\cjna(U)}en.

Prom (5.40), we have g(DTU,Du(^))=g[DuT,-

imi2 i

mi

;

^•^-mi'

Substituting them into (5.45) yields L"(0)

g(DuU, J t=0

ru

I

+ L W\

d\\T\ du

g(DuT,DuT)

+

WT^RmnjWW dt. (5.46)

We write the R term as WTfRmnjWW

=

g(R(T,U)T,U).

(5.47)

Plugging (5.40) and (5.47) into (5.46) yields L"(0) = g(DuU,

^r)

+ Ito ^TF \9&TU, t=o

7" 1

H0

\\T

DTU)

+ g(R(T, U)T, U)} dt

i fd\\T\\\ dt. !|T|| V du J (5.48)

An Introduction

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to Finsler

Geometry

Definition 5.4.2 Let V and W be two arbitrary vector fields along geodesic a~{t), a < t < b, with velocity T(t). The following quadratic form I(V, W) := j

±r

[g(DTV, DTW) + g(R(T, V)T, W)\ dt

(5.49)

is called the index form along a. Note that in (5.49) g is evaluated at the point (a(t), [T(i]}) of SM. By using (5.47) and the property of Riemannian curvature, we see I(V, W) is indeed symmetric. If the geodesic is given a constant speed parametrization, then we define I(V, W) omitting the factor -n^nr.

Chapter 6

Geometry of Projective Sphere Bundle

The key idea in Finsler geometry is to consider the projective sphere bundle SM on the n-dimensional Finsler manifold. The main reason is that all geometric quantities constructed from Finsler structure are homogeneous of degree zero in y and thus naturally live on SM, even though Finsler structure itself does not. SM is a (2n—l)-dimensional Riemannian manifold with its Sasaki type metric induced by the fundamental tensor. In this chapter, we will give intrinsic relation between several important subbundle, such as Finsler bundle of SM and Finsler manifold itself. In particular, we show that Finsler bundle is integrable if and only if manifold itself has zero flag curvature, and Finsler bundle is minimal if and only if manifold itself is Riemannian. 6.1

Riemannian connection and curvature of projective sphere bundle

Let (M,F) be an n-dimensional Finsler manifold and SM its projective sphere bundle with the Sasaki type metric G. Then (SM, G) is a (2n-l)dimensional Riemannian manifold where G := Sijuj* ® ujj + 6a/3Ljna w / . Throughout this chapter, our index conventions are as follows: 1
k, •••

1 < a,/3,7,---
1,

1 < a, b, c, • • • < In — 1. 69

(6.1)

An Introduction

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to Finsler

Geometry

We introduce the abbreviations ipi = SijUj;

ipa = SaB^n13,

ac = n + a.

(6.2)

Then from (6.1) the metric II can be rewritten as II = ^ a

^<* ® V>o = 5 1 ^ a ® V>a + i>n ® Ipn + ^ 4>& ® V'aa a

(6.3)

From (4.1),(4.2),(4.3) and (6.2), we have cfa/>Q =

dujc wk A Wfcc

(6.4)

dVn = dw™ = wfc A Wfc™ = w'3 A tjpn = - ^2 V>« A i>s,

cfa/'a =

(6.5)

du>na

= nna+ujnkAUka = \ Z)»,j RnaijUi A Wj + E i Pnai0Ui A ( j / + U n ^ A W^ = \RnaB^B AW 7 + ^ RnaBnUB A Wn a

(6.6)

& A

+ E 7 -Pn 7/3^7 A 0/,/ + J2/3 0 "P™ — 2 S/3,7 RaBjSg A # 7 + £]/3 RapU>3 A Wa + E/3, 7 LocB^B A 6>7 + J^/3 ^ A W^, where ii a ; 3 7 := SaeRneg^ and L a / 3 7 is the Landsberg curvature. Note that the length square of Rap, denoted by ||5|| 2 , is a scalar function on SM. Lemma 6.1.1

(M,F) has scalar curvature K if and only if RocB = K6ag.

(6.7)

In particular, (M,F) has zero flag curvature if and only if \\S\\2 vanishes identically.

Proof.

Obviously.

D

Geometry of Projective Sphere

Bundle

71

Let ipab be the Levi-Civita connection 1-form satisfying

d a=

^

~ Yl ^ab A ^6; ^ab+^ba = °-

(6.8)

b

From (6.4) and (6.8) we have

Y2^PA ^"P ~ u ^

+

^n

A

^an

-

V'a) + X ] ^ A i/)ap = 0.

P

(6.9)

/3

From (6.5) and (6.8) we have ^

V'/J A (r/>„/3 + ^ )

+ ^

V^ A T/>„0 = 0.

(6.10)

From (6.6) and (6.8) we get

0

7

7

+ V»n A (V>6„ + X Rap^p) + 5 Z ^J8 A ( ^ ~ ^Z3") = 0/3

/3

(6.11) By (6.9)-(6.11), we have £ b #6 A $ab = 0, where

(6.12)

Wab)

By Cartan's lemma, we have ab

~ / j^abcYci

a

abc = O-acb'

(6.13)

An Introduction

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to Finsler

Geometry

Substituting it into (6.12) yields w

/3a + Z ) aa0ci>c

V'fi + X) aanci>c

a

a0c'4,c

£)

(6.14)

(^a&) =

vrTd

- E ^ ^ 7 +E«^«

+ 2 ^ aa/3cV>c

Put babe =

&abc T" &6ac

(6.15)

Then ^

n/?

^a/JcV'c =

^

= ~ x ( w a / 3 + Wpa) = ^

Haf3^,

from (5.8), (6.12), (6.13) and (6.15). Similarly, we have ^2banc1pc

= -{'dan

+ $na)

= 0,

and

Thus Q>a0c)

' 0 ' 0

J^/3o:7

{Knc) = 0,

o 2Lapy

(bapc) = -

(6.16)

Similarly, we get

(.Oj^cJ

1 7,

'

-tia'y

0

,

[ u J

(bnnc) = 0,

(bagc)

=

0 ' 0 Hap-,

(6.17)

(6.15) together with aQ&c = aacb, suggests that the quantity aabc is algebraically similar to the Christoffel symbols of the first kind in Riemannian geometry, namely {ijk} = \{gijtk — 9jk,i + 9ki,j)- Imitating the usual derivation of the formula for {ijk}, that is, applying (6.16) and (6.17) to

Geometry of Projective Sphere Bundle

73

the combination babc — hca + bcab leads to intermediate formulas for aa&c which, in conjuction with (6.15), imply that

(ipab) =

E/3 ( \RaP -dapWp

°

\E

R 7

Pl^7

(6.18) Let \I/ab denote the curvature form of (SM, II) and let Kabcd be its components. Then dl/jab = -^2lpac^1pcb

+ ^ab;

^>ab = -Z ^Kabcdlpc

2

A V'd-

(6.19)

c,d

Substituting (6.18) into (6.19) yields * r a = C^an + 22 ^ c

A

V'cn

o = ^

Rp1(5ap

~ -7Rap)4>~, A ^ n

4

m o d Va A f^,

V'a A 1p&,

/3,7

(6.20)

* n s = Yl RocpRff^n A V^-

(6-21)

0,7

Comparing (6.19) with (6.20), we have o Kanan = /_^ Rap($ap ~ jRap)-

(6.22)

a,P Comparing (6.19) with (6.21) we get Knuna — T /

,, Rgp•

(6.23)

Denote by I the dual vector of Hilbert form with respect to II (cf.(4.7)).

74

An Introduction

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Geometry

Then the Ricci curvature in the direction of I can be written as Ric(l)

= J2a =

Kanan

2-^a\

anan

'

**-nana)

= Eo,/3 Raf3(5a0 ~ 4-Ra/3) + 4 E Q) /3 Rl/3 = Ric-\\\S\\2. T h e o r e m 6.1.2 ([Mo, 1998]) manifold. Then

Let {M,F)

n—1 Ric(l) < min{—-—,

be an n-dimensional

Ricci

(6.24)

Finsler

scalar}

and n-1 Ric(l) = —-—

(resp.

Ricci

scalar)

if and only if (M, F) has constant flag curvature 1 (resp.O ) . Proof.

Notice that

\\S\\2: =

E^fa(3 (6.25)

iRic2.

= n—1

Plugging it into (6.24) yields Ric(l) < Ric <

2(n - 1)

Ric2

Ric

n-1

n-1

V2(" - 1)

<

n-1

If Ricil) = ^ f i , then (6.25) implies iplies that Rap = 0 if

Rn

a^/3

•"n-l.n-l'

Hence Raa —

Ric = 1, Va. n-1 By using (6.24) we can obtain another conclusion immediately.

•

Geometry of Projective Sphere

6.2

Bundle

75

Integrable condition of Finsler bundle

Let (M, F) be a Finsler manifold and SM its projective sphere bundle. We study the horizontal subbundle H of SM. By (3.4) H := {b e TSM\0&(b) = 0}. Because H is isomorphic to Finsler bundle p*TM (cf. Section 1.4), we call H the Finsler bundle of (M, F). In this section we will investigate the integrability of Finlser bundle Ti. Using Probenius' theorem, H is integrable if and only if dipa = 0

mod

ipa-

(6.26)

By using (6.6) d a

^

1

T—i

A

= o z2

Ra

0~(®P A V*7 + z2 RaP^I3 A V'n

/3,7

mod

I/J& ,

/3

and ^ot/3"f

==

^a"ff3

• = = Vae-LLn fl~f

Hence (6.26) holds if and only if Ra0 = 0,

-Ra/37 = 0.

(6.27)

Lemma 6.2.1 Rafif = ^ {Rpa;i — R-ya;f}),

(6.28)

where R/3a-~/ is defined by R^-y^rC

Proof.

= dR0a + J V w 7 a - RyaLJ^

mod

w*.

Review the curvature 2-form £V is defined by £V =dajij -u>ikAcjkj.

(6.29)

Differentiating it, one deduces the following second Bianchi identity dfV = -fii f c A wfcJ' +

fc Wi

A IV'.

An Introduction

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Geometry

In particular, taking i = n, j = a, we get

dnna + nj A u^ - a0a A uj = o.

(6.30)

Plugging (5.4) (i = n, j = a) and its differential into (6.30), yields -RQ/3;7 = -R a /37 + -R 7 Q /3n + L" f3j\n-

(6.31)

On the other hand, using the First Bianchi Identity (5.6) we have Ra/3-fn + R-ff3na + Rn/3a~f = 0>

(6.32)

From (6.31) we get = 5p£[R^e.a — R€~/a — £ e 7 a|n] =

-ft7 ;a

(6.33)

-^-/a/?7|n'

-ft/370;

Similarly, *^y[3na —

-^7/3 cm

= — (-"a =

xla

;7 ~ #/3a- 7 — £ 7 / 3 a | n ) j-y

(6.34)

-K/37CI T -^Q/37|n-

Plugging (6.33) and (6.34) into (6.32) yields U = Kry p

;Q.

/3

— Kp~fa — La/)y\n _

p

0

+ {—Ra

;7

— lifj^a

T ^a/37|n) + Rn/3a-f

_QD„

n Theorem 6.2.2 ([Mo, 1998]) l e i (M,F) be a Finsler manifold. Then its Finsler bundle H is integrable if and only if (M, F) has zero flag curvature. In particular, both Riemannian manifold and locally Minkowskian manifold have integrable Finsler bundle.

Proof. Suppose that the Finsler manifold (M, F) has vanished flag curvature. Then -R7/gQ = 0 from Lemma 6.2.1. Thus (6.27) holds and the Finsler bundle H is integrable. The necessity of Theorem 6.2.2 can be obtained from (6.26) and (6.27) immediately. •

Geometry of Projective Sphere

Bundle

77

Now we consider the vertical subbundle V of SM, i.e. the projective spheres {SXM} (cf. Section 3.4). V is determined by #j = 0. Restricting (6.18) to V, we have

7

4>na = 0.

Hence the components of the second fundamental forms of the immersion SXM^SM are h% = -LaM;

hn^ = 0.

(6.35)

Using Sasaki-type metric G, one decomposes the Finsler bundle H into direct sum as follows H = Hi ©spanjj}. From (6.35) it is easy to show the following Proposition 6.2.3

The first normal spaces of projective spheres lie in

HL

Remark (M, F) has flat horizontal foliations if and only if it has zero Riemannian curvature. 6.3

Minimal condition of Finsler bundle

Let V now denote an arbitrary smooth distribution on a smooth Riemannian manifold (N, h), which we call a vertical distribution. Let H be the distribution orthogonal to V, we call this a horizonal distribution. Then TN = H®V. Definition 6.3.1 V is said to be Riemannian if Cuh(X,Y) = 0 for all x £ N, U G Vx and X, Y G Hx, where L\j denotes the Lie derivation with respect to U. Remark If V is integrable then V is Riemannian if and only if its leaves are locally the fibres of a Riemannian submersion. In this case, we say that the foliation determined by V is Riemannian.

78

An Introduction

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Geometry

Let V denotes the Levi-Civita connection for h and let V denote the orthogonal projection onto V. By the second fundamental form of H at x € N we mean the symmetric bilinear form C = Cx'•Hx x Hx

(X,Y)->±V(VXY

V,

+

VYX)

(where, to evaluate the covariant derivative, X, Y are extended to local vector fields ). When ||X|| = 1, C{X,X) is called the normal curvature of H in the direction X. The vector dii ^ v Traced is called mean curvature of H at x. The Finsler bundle H is called minimal (resp. totally geodesic) if its mean curvature (resp. its second funamental form) vanishes. Now we characterize Riemannian manifolds in terms of their Finsler bundles. Using the metric II and its orthogonal coframe {V'o}, we define an isomorphism \i: Hf —> V by

u((i(e),x) = [\(e)}(x). where A : HI —> V* is given by

Lemma 6.3.2 Indentifying H{ with V via \i, the second fundamental form of the Finsler bundle is just the same as the Cartan tensor of (M, F). In particular, the mean curvature of the Finsler bundle is equal to the Cartan form.

Proof.

From (6.18), we have "4>Pa = ^ ( J 4 Q / 3 7 ) V ' 7 - -Ra/ltl'n

V'™ =

-y^Rgpipp.

m o d Vfi,

(6.36)

(6.37)

Geometry of Projective Sphere Bundle

79

Denote the second fundamental form of the Finsler bundle H by £. Then its components £ a satisfy that
<8> V'a + V"i ® Vte]

m o d Va

(6.38)

= (/) + (ii), where 2(7) : = Y,p[i>pa ® i>p + i>p Q Vvsa]

mod -0a

= S / j[E 7 (H' Q , / 3 7 + -Rayp)lp-y - ^Rap1pn] ® V"/3 1 1 +2/3^/3 ® [ S 7 ( F a / 3 7 + -Rayp)l/>^ -Raplf)n] = ^0n(HaPi + Hajp)ipp <8> ikf + | S / 3 i 7 ( J R a / 3 7 + Raip)ipp <8> ^ 7 -^pRap{^n

(6.39)

®1pp+1pp® lf>n)

- 2 £ / 3 , 7 # a / J 7 ^ / ? ® ^ 7 - (VW ® Ipn + Ipn <S> Ipna)

= 2Y,pnHQp7ipp ® V7 - 2(11).

p.

With the identification # f ~ V, we get C-^2

^&^a

=

^2 i'a ip-y = A .

In particular, r? := ^trC ~ £ t r A Combining Theorem 2.2.4 and Lemma 6.3.2, we immediately obtain the following: Theorem 6.3.3 ([Mo, 2000]) A Finsler manifold is Riemannian if and only if its Finsler bundle is minimal. Corollary 6.3.4 A Finsler manifold is a flat Riemannian manifold if and only if its Finsler bundle has minimal leaves. Proof.

Follows from Theorem 6.2.2 and Theorem 6.3.3.

D

80

An Introduction

to Finsler

Geometry

In general, we have: Proposition 6.3.5 ([Wood, 1986]) Suppose V is integrable. Then the foliation determined by V is Riemannian if and only if the corresponding horizontal distribution is totally geodesic. By using this result, we give a unified description of Riemannian, Landsberg and weak Landsberg manifold in terms of the property of projective spheres as follows: Proposition 6.3.6 A Finsler manifold is Riemannian (resp. Landsberg; weak Landsberg) if and only if its all projective spheres are Riemannian (resp. totally geodesic; minimal). Now we consider the Hilbert form. It is an important section in the Finsler bundle which arise from Hilbert's problem 23 of his famous Paris address of 1900. Note that, from (6.38) and (6.39), we have Ca = 0 mod ip/3 for

V

a.

Therefore we obtain the following Proposition 6.3.7 The Hilbert form is an asymptotic direction of Finsler bundles, that is, the normal curvature of I (the dual section of Hilbert form) is vanishing.

Chapter 7

Relation among Three Classes of Invariants

As we see above, there are three classes of geometry invariants on a Finsler manifold. Two classes are non-Riemannian, and one class is Riemannian. Recently, we find some important delicate relation between Riemannian invariants and non-Riemannian invariants. In this chapter, we establish the fundamental relation between tha Cartan tensor, the Landsberg curvature curvature and the flag curvature in spiring of the equation of Akbar-Zadeh in constant curvature manifolds. Then we give the intrinsic relation between the S'-curvature, the flag curvature and the Cartan form by discussing subtle Ricci identities.

7.1

The relation between Cartan tensor and flag curvature

Proposition 7.1.1 ([Mo, 1999]) Let (M,F) be a Finsler manifold and Hija (resp. Raf}) its Cartan (resp. flag curvature) tensor. Then Ha0i + Ha/3eRe-t + 7-R a;7 + T-R"/3;7 + r-R 7 ;a + ~R°'f,(3 = 0

(7.1)

where Hap~{Ujn = dHapy — 2_j Hifj-fUJa — 2_^ Hai^LOf}1 — 2_^ HapiU>J

mod

a/,

81

u>ne.

(7-2)

82

An Introduction to Finsler Geometry

Proof,

Using Lemma 6.2.1, we have RP Rp-fac —

- Rf3 3

,

K<-4)

where Rp^a := 5j3eRne-ya and R^ka is the Riemannian curvature of (AT, F). On the other hand, from (6.38), we get + Rya/3n,

(7.4)

where Lapf

'•= —Ha/3y

is the Landsberg curvature of (M,F). (7.4) we obtain that r? 0 — D/3 _ ^ c * ^ n — • fx 7;c*

D/3 Jrij

(7.5)

Substituting (7.3) and (7.5) into _

7a

= R i;a + a(R Q;7

r ^afi^ln _

R y;a) + Hap7

(7.6)

- 2 D/3 , 1 D/3 , rx a — 3 7! a ' 3 t 7 "^ • / J ap7)

together with (5.13) this gives — 2 D/3 1 I D^ 1 i> a — 3 7 » a "* 3 » 7 ' -"01/37

+ gi? a 7 ; i a + 3i? a /3 ; 7 + i?/3Q7 + 2Hap<:Rey — 2(-Oa/37 + HaptRe~t + g i t p a ; 7 + gRapn + jjit p 7 ; a + g i l " ^ ) ,

_

In particular, when M has constant flag curvature, then #>

= 0,

(7.7)

R% = c(5£7,

(7.8)

7

we get a second-order differential equation for the Cartan tensor as follows (see [Akbar-Zadeh, 1988]) tfQ/37 4- cHa07 = 0.

(7.9)

Furthermore, if M has Landsberg type (for details see Definition 4.3.1) and c ^ 0, then M has vanishing Cartan tensor. Hence we obtain Theorem 7.1.2 ([Akbar-Zadeh, 1988]) Suppose that {M,F) is a Landsberg space with non-zero constant flag curvature. Then (M, F) is a Riemann space with non-zero constant sectional curvature.

Relation among

7.2

Invariants

83

Ricci identities

In this section we give some important Ricci identities for a Finsler space which will be used in later. Let (M, F) be an n-dimensional Finsler manifold and SM its projective sphere bundle. Let / : SM —> R be a smooth function. From (4.4), we have df = f\aS + f-,*ujna,

(7.10)

where \i (resp.;a) denote horizontal (resp. vertical) covariant derivative. Differenting (7.10) and using the sructure equations one deduces that Df{i Aw' + Df.a Aio n a + f.iaflna

= 0,

(7.11)

where Af|i = # | ; - / b W ,

(7.12)

Df-a = df.,a - f;0LjJ-

(7.13)

Recall that the cuevature 2-form fia6 have the following structure W

= \Ri3kluk

AUJ1+

Pijkauk

A tona.

(7.14)

Substituting (7.12), (7.13) and (7.14) into (7.11) we have f\Hi — f\j\i + f;ocRnaij,

(7-15)

f\i;a = f;a\i + f^PJic,

(7.16)

/;a;/3 = f;0;a-

(7.17)

We denote simply / | n by / because it is a globally defined function on SM. Let {ej, e a } be the dual basis of {uJi,ojna} with respect to G. By (7.10) and (7.12) we have f\n;a = (Df\n)(es) = (df\n - f\bUjn )(e a ) = f-a - /| Q .

(7.18)

Note that p P

-

TP

-

irP

- n

(7.19)

84

An Introduction

to Finsler

Geometry

Together with (7.16) and (7.18) we get / ; a | n = f;a ~ f\a-

(7-20)

By (7.10) and (7.12) we have f\n\a

= {df\n - /|jW„ J )(e Q ) = f\a.

= Df\n{ea)

(7.21)

Differentiate (7.20) along the Hilbert form and use (7.15),(7.16) and (7.21). We have J;a\n\n

J;a\n =

J\a\n

J\n;a ~ J\n\a ~ J;P-"- a

= f\n;a ~ f\a ~ f;$R

a-

Together with (7.20), we have the following Lemma 7.2.1

For any smooth function f on SM, we have f;a\n

f;a\n\n

= f;a ~ f\a,

(7.22)

= f\n;a ~ f\a ~ ftfR

a-

(7.23)

Corollary 7.2.2 Let (M,F) be a Finsler manifold. Then its Cartan form r)a, S-curvature S and Flag curvature tensor R^a satisfy

(?L^l Proof.

=ii

+

" ^

(724)

-

Recall that T-a = T]a;

where r is the distortion of (M,F). Va

=

(7.25)

By using (7.23) and (7.25) we have

7";a|n|n

~ ''"In;a =

T|„ = — ,

^~\a

7

";/3-*^ oc

\~p)\n;a ~~ \~p)\u ~ V/3-K- a-

r-.

Relation among

7.3

Invariants

85

The relation between S-curvature and flag curvature

Lemma 7.3.1 Let (M,F) be a Finsler manifold. Then rja + VpR13* + \{RiC;<* + 2R0a;0)

Proof.

= 0.

Contract (7.1).

(7.26)

•

Corollary 7.3.2(Z. Shen) Suppose that (M, F) is a Finsler manifold with constant curvature c. Then ria + cr/a = 0.

(7.27)

Furthermore, if F has weak Landsberg type and c ^ 0. Then (M,F) is a Riemannian manifold with non-zero constant sectional curvature. Proof.

Suppose that (M, F) is of constant curvature c. Then Ric]a = 0.

Together with (7.7), (7.8) we have (7.27). If F has vanishing mean Landsberg curvature, then Ja = —r]a = 0. Combine with c ^ O w e get r\a = 0. By using Deicke's result we have (M, F) is Riemannian (cf. Theorem 2.2.4).

• Theorem 7.3.3 Let (M, F) be a Finsler space. Then its flag curvature tensor R^a, S-curvature S, Ricci scalar Ric and Cartan form r}a satisfy i]a + VpR^c, = (j)

Proof. 7.4

~ (j)

= -\{Ric-,a

+ 2i?/3Q;/3).

Combine Corollary 7.2.2 with Lemma 7.3.1.

(7.28)

•

Finsler manifolds with constant S-curvature

Recall that a Finsler metric is called to be constant .S-curvature if S = (n + l)cF for some constant c, where n = dim M. Now we consider a Finsler manifold (M,F) with constant s-curvature. By using (7.24) we have i)oL + -npR^a = 0.

(7.29)

An Introduction

86

to Finsler

Geometry

This equation deduce the following Theorem 7.4.1 ([Shen, 2001]) Let (M,F) be a compact Finsler manifold with constant S- curvature . If F has non-positive flag curvature, then F has weak Landherg type. In particular if its flag curvature is negative at x € M, then F is Riemannian at x, i.e. F2(x,y) = ]T\ . gij(x)ylyJ. Denote the principal curvature of (M, F) by «i, • • • , K „ _ I , i.e. Ra@ = KaSa/3.

(7.30)

If (M, F) is a weak Landsberg manifold with constant S-curvature, then we get rjaKa = 0,

a — 1, • • • , n — 1.

Hence we get Theorem 7.4.2 ([Mo, 2005]) Let (M,F) be an n-dimensional weak Landsberg manifold with constant S-curvature. If K\,- • • , « n _i ^ 0, then (M, F) is a Riemannian manifold. Recall that a Finsler manifold is of Berwald type if it has vanishing Minkowskian curvature. It is easy to see locally Minkowski manifolds and Riemannian manifoldas are all Berwald manifolds. Furthermore, Berwald manifolds are of weak Landsberg type and satisfy S — 0[Shen, 1997]. we have following: Corollary 7.4.3 Let (M, F) be an n-dimensional Berwald manifold. Suppose that Ki, • • • , Kn-i ^ 0, then (M, F) is Riemannian. For a Finsler surface, its Gaussian curvature K takes the place of the flag curvature in the general case. Theorem 7.4.4 ([Mo, 2005]) Suppose that (M,F) is a Finsler surface with constant S-curvature. Then the Gauss curvature of(M,F) lives on M. Note that a Finsler surface is locally Minkowski if and only if it has vanishing Gaussian curvature and Minkowskian curvature. Thus our corollary

Relation among

Invariants

87

7.4.4 and Theorem 7.4.5 are a natural extension of Szabo's theorem [Szabo, 1981]. Szabo's result tells us any Berwald surface is Riemannian or locally Minkowskian. In Theorem 7.3.2, the condition of (M, F) to be weak Landsberg can not be removed. In fact Bao and Shen recently constructed, by using Randers metrics, a family of non-Riemannian Finsler structrues Fk on S3 with constant flag curvature one and vanishing S-curvature [Bao and Shen, 2002]. From the well-known formula (7.9) due to Akbar-Zadeh, we see that the Fk are not weak Landsberg structures. According to Corollary 7.3.2 (cf.[Shen, 2001], Theorem 9.1.1]), any weak Landsberg metric of non-zero constant flag curvature is Riemannian. Here we weaken Shen's condition on the constant flag curvature and impose the constancy of 5-curvature on the Finsler metric instead.

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Chapter 8

Finsler Manifolds with Scalar Curvature A Finsler manifold (M, F) is called to be of scalar curvature if for any y € TM\{0} the flag curature K(P,y) is independent of the tangent P containing y. When dimM=2, (M, F) is automatically of scalar curvature. In this chapter we will discuss Finsler manifolds with scalar curvature by using the fundamental equations derived in previous chapter. We particular determine the flag curvature when certain non-Riemannian quantities are isotropic and give Akbar-Zadeh type differential equations of Matsumoto torsion. We also give a series of their applications.

8,1

Finsler manifolds with isotropic S-curvature

In this section we partically detemine the flag curvature of a Finsler manifold when its ^-curvature is isotropic. Theorem 8.1.1 ([Chen, Mo and Shen, 2003]) Let (M,F) be an ndimensional Finsler manifold of scalar curvature n(x,y). Suppose that the S-curvature is isotropic, S=(n+l)c(x)F(x,y), where c{x) is a scalar function on M. Then there is a scalar function o~(x) on M such that K

= 3c 4- <x.

In particular, c(x) = c is a constant if and only if K = K(X) is a scalar function on M. In this case, K = constant when n > 3.

89

An Introduction

90

Proof.

to Finsler

Geometry

Suppose that (M, F) is of scalar curvature K. It is easy to show

n?a = Ks0a,

(8.i)

Ric;a — (n — 1)K ; O,

B13

-

a-

(8.2)

(8.3)

< Plug them into (7.28), we obtan

fS\

fs\

\?L . - ( ? ) 1-

= -i(n

(8.4)

3

Note that

l=(-

• +l)c(x).

Hence 1 ^|n;o:

^|<

(8.5)

By using (7.16) and (7.19) , we have ^-jn;a

^;a|n

=

*-*•

(8.6)

(7.22) tells us

Substituting (8.6) and (8.7) into (8.5) yields (K — 3c);Q, = 0. Thus a := K — 3c is a scalar function on M. In terms of the local coordinat (xz, yz) on M, we have 0c 2/* Thus K = K(X) if and only if c = constant. In this case, K = constant when n > 3 by the Schur theorem (cf. Proposition 5.2.5). •

Finsler Manifolds with Scalar

Curvature

91

For any Finsler manifold (M, F) with positive constant flag curvature and zero S'-curvature, Kim and Yim have showed t h a t if F is reversible, then F is Riemannian. Thus we obtain C o r o l l a r y 8 . 1 . 2 Let (M,F) be an m(> 3)-dimensional reversible Finsler manifold with zero S-curvature and positive scalar curvature. Then (M, F) is Riemannian. We have a lot of Finsler surfaces whose Gauss curvature are not scalar functions (see Section 8.3). In the case t h a t dimM > 3, we also have m a n y examples of Finsler manifolds of scalar curvatures, with a non-trivial dependence on y. Recall t h a t a randers metric a + /3 is locally projectuively flat (and therefor of scalar curvature) if and only if a is of constant curvature and (3 is a closed 1-form. Let (M, a) be a compact Riemannian manifold with constant sectional curvature R = 1 and j3 be an arbitrary closed 1form on M. T h e n the Randers metric Fe — a + e/3 is of scalar curvature with negative curvature for small e. If in addition, K = K(X) is a scalar function on M (in which case it is constant when n > 3 by Schur theorem) then (3 = 0 follows from Akbar-Zadeh's rigidity theorem: every Finsler metric on a compact manifold of negative constant flag curvature must be Riemannian. We can always conclude t h a t the flag curvature K depends on y if n > 3 and j3 / 0. R e m a r k T h e reversibility in Corollary 8.1.2 can not be dropped. Z.Shen has constructed many non-Riemannian Finsler metrics on Sn with constant flag curvature 1 and zero S'-curvature.

8.2

Fundamental equation on Finsler manifolds w i t h scalar curvature

Let (M, F) be a Finsler manifold. Let M^

:= Hal3l

1 1 —-n a 5(3 y

1 —— r/pS^a

-—rj-r^ap

(8.8)

where i? Q / 3 7 is C a r t a n tensor and r\a is C a r t a n form. We obtain a totally symmetric tensor. Note t h a t M^ = 0 for Finsler surfaces. T h e quantity M a / 3 7 is introduced by M.Matsumoto [Matsumoto, 1972].

92

An Introduction

Definition 8.2.1 torsion.

to Finsler

Geometry

We call totally symmetric tensor M a ^ 7 the Matsumoto

Matsumoto proves that every Randers metric satisfies that Map^ = 0. Later on, Matsumoto-Hojo proves that the converse is true too. T h e o r e m 8.2.2 ([Matsumoto and Hojo, 1978]) A Finsler manifold of dimension n > 3 is a Randers manifold if and only if it has zero Matsumoto torsion. Remark The Matsumoto torsion gives a measure of the failure of a Finsler manifold to be a Riemannian manifold. Proposition 8.2.3 ([Mo and Shen, 2005]) Let (M, F) be a Finsler manifold with scalar curvature K. Then its Matsumoto torsion satisfies the following Fundamental Equation: Maf}-y + KMa01 = 0

Proof.

(8.9)

Plugging (8.1),(8.2) and (8.3) into (7.28) yields r/Q + naK = --(n+l)K-a

(8.10)

that is, K.a =

3 —-(f)a + Kr)a).

(8.11)

On the other hand, substitute (8.1),(8.2) and (8.3) into (7.1) we have HaPy + KHapy + -5afjKn + -<5/37K;a + Now (8.9) follows from (8.8), (8.11) and (8.12).

-<57QK;/3

= 0.

(8-12) •

Using the fundamental equation (8.9), Theorem 8.2.2 and ODE theory one can show the following T h e o r e m 8.2.4 ([Mo and Shen, 2005]) Let (M,F) be a compact negatively curved Finsler manifold of dimension n > 3. Suppose that F is of scalar curvature. Then F is a Randers metric.

Finsler Manifolds with Scalar

Curvature

93

Theorem 8.2.4 greatly narrows down the possibility of compact negatively curved Finsler manifolds of scalar curvature. Although Randers metrics are very special Finsler metrics, the classification of Randers metrics of scalar curvature has not been completely done yet. See [Yasuda and Shimada, 1977; Matsumoto, 1989; Matsumoto and Shimada, 2002; Bao and Robles, 2003; Bao, Robles and Shen, 2004] for the classification of Randers metric of constant flag curvature. A Finsler metric is said to be locally projectively flat if at any point there is a local coordinate system in which the geodesies are straight lines as point sets. Based on Theorem 8.2.4, we obtain following Theorem 8.2.5 Let (M, F) be a compact negatively curved Finsler manifold of dimension n > 3. F is locally projectively flat if and only if F = a-\-(3 is a Randers metric where a is of constant sectional curvature and (3 is a closed 1-form. Proof. Since locally projectively flat Finsler metric F has vanishing Weil curvature and Douglas curvature, we get F has scalar curvature. Recall that the Weil curvature gives a measure of the failure of a Finsler metric to be of scalar curvature. By using Theorem 8.2.4, F is a Randers metric. Vanishing Douglas curvature implies that f3 is a closed form and a has vanishing Weyl curvature too. Thus a is of constant sectional curvature. Conversely, if d(3 = 0 then F = a + f3 has vanishing Dougals curvature. Together with a's constant curvature, F has vanishing Weil curvature. Thus F is locally projectively flat. • Corollary 8.2.6 Let (M,F) be a compact negatively curved Finsler manifold of dimension n > 3. Suppose that F is of scalar curvature and weakly Landsbergian. Then F is Riemannian. Proof. It follows from Theorem 8.2.4 that F is a Randers metric. Thus F has vanishing Matsumoto metric from Proposition 8.2.3. Since F is of weak Landsberg type, 0 = M a /3 7 = -WQ/3 7 =

—La0y.

94

An Introduction

to Finsler

Geometry

T h a t is, F is a Landsberg metric. T h e n the follows from N u m a t a ' s Theorem [Numata, 1975]. •

According to N u m a t a [Numata, 1975], any Landsberg metric of scalar curvature with non-zero flag curvature is Riemannian. Corollary 8.2.6 weaken N u m a t a ' s condition on the Landsberg curvature and impose the compactness on the manifold instead. We do not know if Corollary 8.2.8 is still true for weakly Landsberg metric of scalar curvature with positive flag curvature. It is known t h a t every weakly Landsberg metric of non-zero constant flag curvature is Riemannian (see Corollary 7.3.2). Let S be the S'-curvature of a Finsler manifold ( M , F), and let 1 Eij

:=

d2S

2 dyWyi

'

Ey

= Eij

^' ^ ^

®

dx

*

D e f i n i t i o n 8.2.7 We call E := {Ey} mean Berwald curvature. to be weakly Berwaldian if E = 0.

F is said

C o r o l l a r y 8 . 2 . 8 Let (M,F) be a compact negatively curved Finsler manifold of dimension n > 3. Suppose that F is of scalar curvature and weakly Berwaldian. Then F is Riemannian.

Proof. It follows from Theorem 8.2.4 t h a t F is a Randers metric. According [Chen and Shen, 2003] a Randers metric has constant Berwald curvature if and only if it has constant S-curvature. T h u s F has constant S'-curvature. Using Theorem 8.1.1, every Finsler metric of scalar curvature with constant S'-curvature must be of constant flag curvature in dimension greater t h a n two. T h u s the flag curvature of F is a non-zero constant. By Akbar-Zadeh's Rigidity theorem [Akbar-Zadeh, 1988], F must be Riemannian. •

T h e compactness in Corollary 8.2.8 can not be dropped. Example 3.3.3 in [Bao and Robles, 2003] is a weakly Berwaldian Randers metric on S 2 x ( 0 , T ) with constant curvature -1 where r is a positive real number. From the proof of Theorem 8.2.5, a Randers metric F = a + /3 is locally projectively flat if and only of a is of conatant sectional curvature and j3 is closed. In Theorem 8.1.1, we find /3 such t h a t F = a + (3 is locally

Finsler Manifolds with Scalar

Curvature

95

projectively flat with S = (n + l)c(x)F [Chen, Mo and Shen, 2003]. Hence we have completely classified the locally projectively flat Randers metric of isotropic 5-curvature.

8.3

Finsler metrics with relatively isotropic mean Landsberg curvature

Let (M,F) be a Finsler manifold and J (resp. 77) its mean Landsberg curvature (resp. Cartan form). (M,F) is said to be of relatively isotropic mean Landsberg curvature if J = c(x)rj where c = c{x) is a scalar function on M. Theorem 8.3.1 Let (M,F) be an n-dimendional Finsler manifold of scalar curvature. Suppose that J is relatively isotropic J-c(x)rj

=0

(8.13)

where c = c(x) is a C°° scalar function on M. Then the flag curvature K = K(X, y) and the distortion r = T(X, y) satisfy ^~K-a

+ (K + c2-c)T;a

= 0.

(8.14)

Moreover, (a) If c(x) = c is a constant, then there is a scalar function p(x) on M such that K = - c 2 + pe~^fr.

(8.15)

(b) Suppose that F is non-Riemannian on any open subset of M. If K = K(X) is a scalar function on M, then c(x) = c is a constant, in which case 2 K = - c < 0.

Proof.

By assumption (8.13), Ja-crja=0

(8.16)

for Va € {1,2, • • • , n - 1}. Note that (see Section 4.3) Ja = -i)a.

(8-17)

96

An Introduction

to Finsler

Geometry

Plugging (8.17) into (8.13) yields -ja

+ nr)a = - - ( n + l)re ;a .

(8.18)

By using (8.16) and (8.17) we have Ja

— CTja -\- CTja

- crja + c(-Ja) = CT]a - c2rja = (c - c2)rja. Substituting it into (8.18) we obtain (C2 - C)r]a + KT)a + - ( n + l)«;a = 0.

Hence (8.14) holds. Recall that r ; a — r\a. (a) Suppose that c — c(x) is a constant. Then equation (8.14) simplifies to n+ 1 , 9. ——K.a + (re + cA)T-a = 0. This implies that

(logp^)-^

= 0,

where p = (K + c 2 )e"+ T . Thus /? = p{x) is a scalar function on M and (8.15) holds. (b) Suppose that re = K(X) is a scalar function on M. Then (8.14) simplifies to (re + c2 - c)r.a = 0.

(8.19)

We claim that c(x) = c is a constant. If this is false, then there is an open subset U such that dc(x) ^ 0 for any x € U. Clearly, at any x £ U, K(X) ^ -c(x)2 + c(x,y) for almost all y G TXM. By (8.19), r ; „ = ?7a = 0. Thus F is Riemannian on U by Deicke's theorem (cf. Theorem 2.2A). This contradicts our assumption in the theorem. This proves the claim. By (8.15) and (8.17) p{x)T-a = 0.

(8.20)

We claim that p(x) = 0. If this is false, then there is an open subset U such that p{x) ^ 0 for any x £lA. By (8.20), we obtain that T-a = r\a = 0 on U. Thus F is Riemannian on U. This again contradicts the assumption in the theorem. Therefore p(x) = 0. We conclude that re = - c 2 by (8.15). •

Finsler Manifolds with Scalar

Curvature

97

Example Consider the Randers metrics defined in (4.16) and (4.17), It is easy to verify t h a t J Oc

eTja

— U,

where c = c(x) is defined in (4.18). when d i m M = 2, a direct calculation show the Gauss curvature is given by = K

- 5 K 2 - 4 C [ e + ( K 2 + eQ|:r| 2 ] 4[e + (K2 + e()\x\2]2

3(K2 + e f l ( l + CM 2 )** [e + ( « 2 + e()\x\2]2F '

In fact, for any Randers metric F = a + f3, F is of relatively isotropic mean Landsberg curvature if and only if F has isotropic S'-curvature and /? is closed. For a general Finsler metric, (8.13) does not imply t h a t S = (n + l)c(x)F.

(8.21)

Now we combine two conditions (8.13) and (8.21) and prove the following: T h e o r e m 8.3.2 scalar curvature. curvature satisfy

Let (M,F) be an n-dimensional Finsler manifold of Suppose that the S-curvature and the mean Landsberg

S=(n

+ l)cF,

where c = c(x) is a scalar function by

J = crj,

on M.

Then the flag curvature

3c 2 + oK = 3c + a =

(8.22) is given

^2r h fie " + 1 ,

where o~{x) and /J,(X) are scalar function on M. (a) Suppose that F is not Riemannian on any open subset in M. c(x) = c is a constant, then K = —c2, o~{x) = —c2 and fi(x) = 0. (b) If c(x) 7^ constant, then the distortion is given by

T

= ln

%

2

2

[6c + 3(c + o-)J

.

If

(8.23)

v

'

An Introduction

Proof.

to Finsler

Geometry

By Theorem 8.1.1 we have 1

for some scalar function a on M. Plugging it into (8.14) yields !

^ « ; a + ( ^ + C2 + | ) T ; Q .

(8.24)

We obtain [(2K + 3c 2 + a) ^

eT\.a = 0.

(8.25)

Thus there is a scalar function u(x) on M such that 2r K

3C2+(T

= ue «+i

—.

(8.26)

Comparing (8.26) and the formula in Theorem 8.1.1, we obtain c=-l(c2+<x) + |

e

- ^ .

(8.27)

(a) Suppose that c(x) = c is a constant. We claim that u(x) = 0. If it false, then U := {x e M,u(x) ^ 0} ^ . From (8.27) r = T{X) is a scalar function on U, hence F is Riemannian on U. This contradicts the assumption in (a). Now (8.27) is reduced to that cr = —c2 and (8.26) is reduced to that K = — c 2 . (b) If c(x) ^ constant, then /x ^ 0 by (8.27). In this case, we can solve (8.27) for r and obtain (8.23). •

Chapter 9

Harmonic Maps from Finsler Manifolds A Finsler manifold is a Riemannian manifold without the quadratic restiction. In this chaper we introduce the energy function, the Euler-Lagrange operator, and the sress-energy tensor for a smooth map (j> from a Finsler manifold to a Riemannian manifold, we show that <j> is an extremal of the energy function if and only if satisfies the corresponding Euler-Lagrange equation and the fundamental existence theorem of harmonic maps on a Finsler space. We also characterize weak Landsberg manifolds in terms of harmonicity and horizontal conserativity. Using the representation of a tensor field in terms of geodesic coefficients, we construct new examples of harmonic maps from Berwald manifolds which are neither Riemannian nor Minkowskian.

9.1

Some definitions and lemmas

Let (M,F) be an m-dimensional Finsler manifold and u>i,--- ,u>m the adapted frame fields on the dual Finsler bundle (cf. Lemma 3.1.1). Let Wij be chern connection 1-form. Recall that gtj = [\F2\yiyi. From (4.1) it is easy to see that U>l A • • • A LJm

A L0ml

A •• • A

L0m,m-1

is the volume form with respect to the Riemannian metric on SM. denote it by II. The following lemmas will be used later. Lemma 9.1.1

We

If M is a compact Finsler manifold, then for any function 99

An Introduction

100

to Finsler

Geometry

f :SM - • R

/

fll= f dx f

JSM

JM

JSXM

fJdet{9lj)X

v

where dx = dx1 A • • • A dxm and m o d dx3 .

X = (jJml A • • • A U>m,rn-1

In particular, if M is Riemannian and f is defined on M, then

fU = VoliS™-1) [ fdv,

[ JSM

JM

where Vol(5 m _ 1 ) is the volume of stand (m — l)-dimendional Proof.

sphere.

Obvious.

•

In this chapter, we set

1 < A, /z, r, • • • < m — 1,

A= m+A

1 < a, b, c, • • • < 2m — 1,

From (4.2) and (4.3), the first structure equation (M,F) can be written as dh)i = ^ w ; - A Wji,

uji:j + ujji = - 2 ^

Hij\vm\,

(9.1)

where Hi^x — A(ei,ej,e\). By using (5.4) the curvature 2-form ttij := du>ij — ^2 uiik A u>kj can be expressed in the form &ij = -Z /_^ Rijkl^k

where Rijki = -Rijik.

A0Ji + 2_^ Pijk\Uk

A 0Jm\,

Set ^Xfiy

'=

-fmA/17

is the Landsberg curvature. From the proof of Proposition 5.1.2, we have 2_^L\\IJ,

x

= - y ^xx^, x

Imi/i = 0,

(9.2)

Harmonic Maps from Finsler

Manifolds

101

where the dot denotes the convariant derivative along the Hilbert form as usual. Denote the Riemannian metric on SM by II. The divergence of a form ty on SM with respect to II is defined by divtt := £ ( £ « „ ¥ ) (

)Co)

a

where {e a } is the dual basis of {wi, • • • ,u>m,uJmi, • • • ,^m,m-i} on T(SM) and D€a is the covariant derivative induced by G along ea. Lemma 9.1.2 (i) ForS = J2 SoJi € T(p*T*M), divS = £ % + £ S M L A A M . (ii)ForT = Y^TijUiUj 6 T(Q2p*T*M),

divT(ei) = ^Tijy

Proof.

+

^T^Lxxn-

With the abbreviations Ipi = Wi,

Ipx =

w

mA

denote the Levi-Civita connection with respect to {i/v} by {tpab}- By using (6.18), we have i>ij=Wji

mod VA,

fe

= ~ "5Z Li\ni>p.

modipj.

Thus divS =

Ea(DSa)(ea)

= E ( ^ i - E s J w iO( e i) - E ^ C - ^ A A M ) = E •S'ili + E Sfj.Lxxv.-i where the covariant derivative of S is defined by DSi = dSi — 2_l SjWij = 22 Si\jui + z2 ^;AwmAThis proves (i). Similarly, for T we have divT( £i ) = £(
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Geometry

which proves (ii).

•

The energy density of a map <j> : (M, F) —> (N, h) from a Finsler manifold to a Riemannian manifold is the function e{) : SM —> M.+ U {0} defined by e(cf>)(x,[y}) = l^hfae^faej),

(9.3)

3

where {ej} is the orthonormal basis with respect to g (the fundamental tensor of F) at (x, [y]). If Q is a compact domain in M, we use the canonical volume element II associated with F to define the energy of : (fi, F) —* (N, h) by

E^,n) = - f e(^)n, where c := Vol(5 m _ 1 ) is the volume of the standard (m — l)-dimensional sphere and SSI, the projective sphere bundle of 0. If M is compact, we write E(<j>) =E((/>,M). Remark By Lemma 9.1, our notion of energy reduces to the usual notion of energy if M is a compact Riemannian manifold. A smooth map <j> : (M, F) —> (TV, h) from a Finsler manifold to a Riemannian manifold is said to be harmonic if it is an extremal of the restriction of E on every compact subdomain of (M, F).

9.2

The first variation

Let (M, F) be a smooth Finsler manifold and g the fundermental tensor of F. Let (N, h) be a Riemannain manifold. Let cf> : (M, F) —> (TV, h) be a smooth map. Set fc = £ < £ e r ( e 2 T ' J V ) ,

l
(9.4)

The first structure equation for (N, h) is d0a = ] T fy A fya, 0Q/3 + 0/3a = 0.

(9.5)

Harmonic Maps from Finsler Manifolds

103

A vector field v along (j> determines a variation t by 4>t(x) =

expHx)[tv(x)],

where i g / : = (—e,e) for some e > 0. Noting that
(9.6)

It follows that

# W = #(£<£)

,97)

Since {a} is the dual frame field of {w,}, from (9.3) and (9.7) we obtain 2e(t) = ^ ( ^ a a j a a f c w : , a ; f c ) ( e i , e j ) = ^ a j ^ . If u has compact support D c M , then c-±-E{t,n)\t=0= [ V a a i ^ | t = 0 n . «' ./ SAf — at Define $ : M x I -> AT by

(9.8)

(x,t) -^Mx)It is easy to see that

Put $*6>Q = <£*0a + badt,

(9.9)

$*#a/3 = ftOcfi + Bcfidt.

(9.10)

Then ^2bava\t=o := b is the deformation vector field, where {va} is the dual frame field of 0a, snd Bap satisfies Baf) = -BPa.

(9.11)

104

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Geometry

Using (9.5), (9.6), (9.9) and (9.10), we obtain d($*6a) =

F(dOa)

= S ( S a / 3 i w i + bpdt) A ($6pa

+ Bpadt).

On the other hand, from (9.6) and (9.9), we have d($*Qa) = dlY.a.aiWi + badt] = Yl(daai) A tOi + Yl aaiduji + dba A dt = Y^idsmO-ai

+ ^t^dt)

(9.13)

A U>i + X) a a i d W j + dSMba

A dt.

Comparing the coefficients of dt in (9.12) and (9.13), we obtain

^

"gf1^

~ dsMba

= ^

bpfit^Pa

- ^2

BpctdpiUJi

I .

(9.14)

Define the covariant derivative of {ba} by Dba = dsMba - Y,bpa;\Um\-

(9.15)

Substituting (9.15) into (9.14) we obtain da, dt

= ba\i-y^Bpaapi,

ba-:\ = 0.

From (9.8), (9.11) and (9.16) we have JJ2aai(ba\i-EB^af}i)Il

c • £tE(cf>t)\t=0 = =

f(J2aaiba[i

a

—

&

J2aaiB0aa0i)n-

= J"Ei(E m a)|in- /E a «i|i^n, where aai\j •— [daai - 2jaQfcWjfc - y ^ ^ * and

X^(5Za™6«)wi = (dcj),b)

6ap\(ej)

(9.16)

Harmonic

Maps from Finsler

Manifolds

is a global section on the dual Finsler bundle p*T*M. Lemma 9.2 we get cftE{4>t)\t=0

105

Using (9.2) and

= /SMdiv(#,6)n-/SM(r(^),fe)n

where T(4) := -(d(j>,r))+trVd 6 T((cj>oPyTN)

(9.17)

and r\ (resp. Vd<^) denotes the Cartan form (resp. the second fundamental) form of <j). The field T(4>) is called the tension field of cf). Theorem 9.2.1 Let be a smooth map from a Finsler manifold M to a Riemannian manifold N. Then is harmonic if and only if it has vanishing tension field. The basic problem for harmonic maps can be formulated in the following manner: Let UQ : M —» N be a map of Riemannian manifolds. Can UQ be deformed into a harmonic map u : M —> N? Assuming that M and N are compact and without boundary, J. Eells and J. H. Sampson have a positive answer when the Riemannian sectional curvature of N is non-positive. This is so-called the fundamental existence theorem for harmonic maps. Let (j> : (M, F) —> (N, h) be a smooth map from a Finsler manifold to a Riemannian manifold. Shen and Zhang has obtained the second variation formula of this energy functional [Shen and Zhang, 2003]. On the 2000' Finsler geometry workshop Professor S. S. Chern conjectured the fundamental existence theorem of harmonic maps on a Finsler space is true. Recently we have proved the fundamental existence theorem of harmonic maps on a Finsler space. Precisely we show the following: Theorem 9.2.2 ([Mo and Yang, 2004]) Let (M, F) be a compact Finsler space and (N, h) be a compact Riemannian manifold with non-positive sectional curvature. Then any map <j) '• (M, F) —> (N, h) is hornotopic to a harmonic map which has minimum energy in its homotopy class.

Let us now express the tensor field in local coordinates (xl) on M and (u ) on N. We denote by gij and Mr*-fc the components of the fundamental tensor and the Christoffel symbols of the Chern connection on (M, F ) , and a

An Introduction

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to Finsler

Geometry

by hap and NTp the corresponding objects on (N,h). Note that NT^y are just the christoffel symbols of Levi-Civita on (N, h) because ha0 are Riemannian. Let V denote the covariant differentiation (of sections of tensor product of p*TM and p*T*M) on SM with respect to the Chern connection. Then, by (2.46) and (2.47a) in [Bao and Chern, 1996], we have

*£-"*>*>*£*' where

^1,=^%,

(9.18)

d \ 1 dgjk "I* 4 (% - ^dgu +, ?§ V V jfc, - Mfc, 9lj

+

M, fc ),

(9.19)

and G' are the geodesic coefficients of (M,F) (cf. Definition 3.3.8). Using the Leibniz rule, we obtain (cf. [Bao, Chern and Shen, 2000].p.41) Vdxi = -MFkldxk

® dxl.

(9.21)

Suppose that 4> '• {M, F) —> (iV, h) is a smooth map. Locally, we can write <j> = (") where each 4>a is a smooth function defined on open subset in M. Let V denote the covariant differentiation on p*T*M {<j> op)*TN. Then (cf. [Eells and Lemaire, 1983])

- 9ija-L du"

+

fj vd/dxi

•r (PJUX \ d/dxi

g^s-,

where dxi: Now

TlJ

datdx*'

ax

du"

Harmonic Maps from Finsler Manifolds

107

and

so that Va/e,.(d
+ "r^^JJdx^, du

where we have used the fact (cf. [Bao and Chern, 1996])

It follows that the components of the second fundamental form Dd(f> satisfy (Vd0)?. = <j>% - MY%4>ak +

N

Y^4>1

Now consider a smooth function / denned on an open subset in M. Set fj

hj

~dxi'

~dxi

and V d/dx t(df) = (Vdf)ijdx>. Then df = fjdx3, (V#)„ = (V4f)(^j, ^ j ) = /« - MT%fk. Thus (9.17) reduces to r(f) = =

-(df,r,)+TrVdf gij[fa-MYkijfk-UJ),

(9.22)

t>=v(£j).

(9-23)

where

Suppose that (/> : (M, i*1) —> (N, h) is a smooth map. By (9.22) we have r ( ^ ) = g^%

-

M

T%<& - i^\.

(9.24)

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Geometry

Hence the tension field of 6 is r£ = d « Q ( - ( ^ , 77) + T r V # )

9ij [-M? + 4>?j ~ MT^t

+ " r ^ f 0J]

(9.25)

TiW+gWT^tffi, A direct calculation using (9.2), (9.18),(9.19) and (9.20) yields (cf. [Bao, Chern and Shen, 2000], (3.3.3))

and

«..

ki

9.3

\dxk

.(j..?z±\ 1vy/^r). dy dy* k

Composition properties

Let 4> '• (M, F) —* (JV, h) be a smooth map, Define on M (0a,d(f>) = ^

aaioJi

where 8a is an orthonormal coframe of h. Then d{J2i a-ai^i) = d((j)*ea) = (f>*d9a

= f(E«aAW = Ef^Af«?a

(9.26)

by (9.5), consider (9.26) as a two-form defined on the projective sphere bundle SM. We have d

(12i

a

aiUi)

= J2

da

<xi A W i + E

daiduJi

= ]T daai Au>i + J2 aaioj A Wji =

YiAc/)*0fa-

It follows that ^ J } a c , j A w , = 0,

(9.27)

where Daai := daai - J2aaju!ij

+ Yla0i4>*8t3a

(9.28)

Harmonic Maps from Finsler

Manifolds

109

Substituting (9.28) into (9.27) yields the following result. Proposition 9.3.1 satisfies

The second fundamental form of <j> : (M,F) -+ {N,g)

a

ai\j

= aaj\i

an

d

a-ai;\ — 0.

Denote the second fundamental form of <j> by Vd(f> = Y,ita(Daai)LUiVa, where va is the dual frame of 6a. It is easy to see that (9.28) is equivalent to (Vd<£)(ej, ej) = V<j,.ei(<j>*ej) -

4)*{Veiej),

where e, is the dual frame of Wj and the last V denotes covariant differentiation (of sections of tensor products of TT*TM and TT*T*M), on SM, relative to the connection wy. Combine with (9.28) we have (VdfiXX, Y) = V^X{^Y) - MVxY) = (W
(9.29)

€T(TT*TM).

Proposition 9.3.2 7/ (M, F) is a Finsler manifold, (N, h) and (P, k) are two Riemannian manifolds and £ C(M, N), ip £ C(N, P), then r(V> oct>) = dij)o T{4>) + trVdip(d(f>, dcp).

Proof.

By using (9.29) we have

Vd(iP o 4>)(X, Y) = V ( ^), X [V> o 4>)*Y] - (V o 4>)*(VxY) = V^o^.xKV' o )*Y] - MV*.x.Y) +MV*.x4>.Y)-U>°MVxY) = (Vd^)(,X, 0.Y) +M^4>.x{*Y)-^(VxY)} = Vdil>(d, d)(X, Y) + di/jo V # ( X , Y).

(9.30)

110

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Geometry

Put d4> = ££ Q ^a where {va} is the dual frame field of {#<*}, then dtp(< d, 77 >) = dip(< T,£ava, 77 >) = E # ( < £ a , i] > va) = E < & , r) > # ( t / a ) = E < £ad4>(va), 77 > — E < dljj^aVa), f) > = < dip O d(f>, f) > .

(9.32)

Taking traces for (9.31) and using (9.17) and (9.32) yields r(V> ° ^) = trVd(^ o ^>)— < d(r/i o <£), 77 > = CZT/> o (trVd) + trV dip (d<j>, dcp) -dtp(< dcp, r) >) = dtp o T ( 0 ) + tvVdip{d).

A smooth map ^ : (M, F) —* (N, h) from a Finsler space to a Riemannian space is said to be totally geodesic if it has vanishing second fundamental form. Corollary 9.3.3 Lei (M, F) be a Finsler manifold, (N, h) and (P, k) two Riemannian manifolds and

P for the composition of

) is the orthogonal projection of T ( $ ) onto ((j) o TT)*TN. More precisely, T ( $ ) = T((/>) + traceP(d(p, d(f>), where /3 is the second fundamental form of i. In particular, we get Proposition 9.3.4 Let : (M, F) —> (Af, /i) 6e a smooth map. Then is harmonic if and only if T ( $ ) lies in the normal bundle of N in P where $ = i o (p. The following result is a natural generalization of Takahashi's Theorem:

Harmonic

Maps from Finsler

Manifolds

111

Proposition 9.3.5 Let i ; Sn —> Rn+l be the standard inclusion. A map : (M, F) —> (Sn, i*(^s^„ + i)) is harmonic if and only if T ( $ ) = -2e()$ where e(4>) is the energy density of . Proof. Because i is an isometric immersion, Proposition 3.3 tells us that is harmonic if and only if r(3>) = A<& for some function A : SM —> .R. Notice that i is a totally umbilical isometric immersion with constant mean curvature one, then t r V d i ( # , d(j>) = tr*hH = -2e(
•

Say that a function & defined on an open set U of (M, F) is subharmonic if T(/C) > 0 at every point on SU. Theorem 9.3.6 4 map : (M, F ) —> (TV, h) is harmonic if and only if it carries germs of convex functions to germs of subharmonic functions. Proof. For k a function on an open set U of N, we shall start from the composition law r(k o ) = dk o r() + traceVdk(d(f>, d). If is harmonic, we get r(k o ) > 0 on 50 _ 1 (C/) so that ko is subharmonic. Conversely, if at a point (x0, [j/o]) £ SM, T(4>)(X0, [y0]) = i u ^ O . Because (N, h) is a Riemannian manifold, we can choose normal coordinates (va) near 4>(XQ). Define a function in the normal coordinates k = bava + T,(va)2

(9.33)

where {ba} satisfy that Zbawa < -4e(4>)(x0, [yo]), wa = dva(w),

Hess(k)\^Xo)

= 2h. (9.34)

Then A; is a convex function near (j>(xo). Using (3.4) and (3.5) we have dk{w)\j,(Xo) = T,bawa. This gives r{k o 4>){x0t [yo]) = dk o T{4>) + tvVdk(dcf), d(j>) = dk(w) + 4e() < 0

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Geometry

contradicting the hypothesis.

9.4

•

The stress-energy tensor

Let <j>: (M, F) —> (JV, h) be a smooth map from a Finsler manifold (M, F) to a Riemannian manifold (N,h). The stress-energy tensor S^ is a tensor on SM denned by S$ : = e{4>)g - 4>*h

where e(<j>) (resp. g) denotes the energy density (resp. the fundamental tensor) of 4> and * h denotes the pull back of the tensor h to a tensor on SM. We say the S^ is horizontally divergence-free if E2=i(-^e;>|SV)(€t>^0 = 0 for all Y € Hp, where {e,} is any orthonormal basis for for the horizontal space Hp and Hp := {X £ TpSM,um\(X)} - 0 (see Section 3.4). Denote the stress-energy S^ of by Sj, = JJSV/u,

®UJJ,

then Sij = e(4>)5ij - 2 j a a i a « j ,

(9.35)

where e() is the energy density of . Then the horizontal divergence of S<j, is divS^ = J2 SijjUi

0'aiO'an}L\Xp}u)i = J2ie(4>)\i

- Ylaai\3aaj

+ e((f))L\\i

^

^

J2aaiaaj\j

Yjaaiaa^LxXfj.}0Ji

-

= ~ Z ) i E amaaj\j

= ~{r{4>),d4>)

—

+ YJ aaiaa^Lx\fj\u>i

+ e{<j>) ^

LxXfi^^

-e()f),

where f\ denotes the covariant derivative of the Cartan form along the Hilbert form, and where we have used (9.2) and (ii) of Lemma 9.1.2. Recall that a Finsler manifold is said to be of weak Landsberg type if f\ = 0 (see Definition 4.3.3). The following theorems are immediate consequences of (9.36).

Harmonic

Maps from Finsler

Manifolds

113

Theorem 9.4.1 Let <j> : (M, F) —> (N, h) be a non-constant harmonic map from a Finsler manifold to a Riemannian manifold. Then Sj, is horizontally divergence-free if and only if (M, F) is of weak Landsberg type. Combing this with Theorem 4.3.4 we obtain the following Wood type result (cf. [Wood, 1986], Theorem 2.9). Theorem 9.4.2 Let (f>: {M,F) —> (N,h) be a submersion from a Finsler manifold to a Riemannian manifold. Then any two of the following condition imply the third condition: (i) is harmonic; (ii) 5^ is horizontally divergence-free; (iii) p : SM —> M has minimal fibres. Lemma 9.4.3 Let : (M, F) —> (N, h) be a smooth map and $ a smooth section on p*T*M. Then div(e(^)*) = div(E(*, F9a)*ea) + *), where div-H is the horizontal divergence. Proof. div (e($tf) = E e ^ i j t f j + e(
(9.38)

by using Lemma 9.1.2 in [10]. On the other hand, div(E(#, 4>*6a)<j>*0a) = div(Y,^jaajaaiuJi) = ~S(^>jaajaai)^ + 'S^fjaajaa:l,Lx\l/ = E*j\ia a ja a i + ^^>-jaQj^aai +S^ , J a Q ja Q ,j|j + £\I> jaajaavL\\v = ^^i\jO,ajaai

+

(9.39)

Y;fyjaai\jaai

+E^jaajTa(). Put = ^ (e((p)5ij - T,aaiaaj) ty^ = e(0)E\I>^j T,^i^aajaai

(9.40)

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(9.38)-(9.40) implies (9.37).

•

Proposition 9.4.4 If suppty (supporting set of ty) is compact, then

[ (divH5^, *)n + f (s+, D*)n = o JSM

Proof.

JSM

Integrating both sides of (9.3.7) and using Green's theorem.

•

We denote the angle between 2-tensor <£ and ^ on SM by £($, $?). By using Theorem 4.4 in [Mo, 200l] we have Corollary 9.4.5 Let (ft : (M, F) —• (N, h) be a smooth map from a closed Finsler manifold. Then C(S+, DdivnS*)

> |,

where equality holds if and only if u is horizontally divergence-free. In particular, if F is of weak Landsberg type, then equality holds if and only if <j> is harmonic.

9.5

Harmonicity of the identity map

In this section we present the harmonic equation of the identity map from a Finsler manifold to a Riemannian manifold in terms of their geodesic coefficients, and we construct harmonic maps from Berwald manifolds which are neither Riemannian nor Minkowskian to Riemannian manifolds. Let / : (M, F) —> (M, h) be the identity map from a Finsler manifold (M, F) to a Riemannian manifold (M, h). As usual we put I = {F) : U(c M) -> E, where locally P(x1,- •• ,xm) — x1. It follows that dF Using (9.24) and (9.25) we have k

T(i

) = -V^r*. - 5<%-

Harmonic Maps from Finsler

Manifolds

115

and hence T N ^ I ^ - ^ ] - ^ .

(9.4i)

Denote the geodesic coefficients of (M, F) and (N, h) by FGl and hGi respectively. By ([Bao, Chern and Shen, 2000], (3.8.3)) we have - ( Gl)yjyk =

T)k

+Hljk,

where Hx jk is the covariant derivative of the Cartan tensor along the Hilbert form. It follows that 5 « r * + gk% = g*[\{FG%iyi = yi{FGk)v>vi

- ff* ] + gkiHt -Hk+Hk = yv(FG%,yl,

^Ai]

where in the second step we used (cf. [Bao, Chern and Shen, 2000], (2.5.11)). Similarly, for the Riemannian metric h we have \(FG%jyk=hrjk.

(9.43)

Substituting (9.42) and (9.43) into the harmonic equation (9.31) gives T? = \gi>(hGk-FGk)yiyi.

(9.44)

Thus we have the following result. Proposition 9.5.1 Let (M, h) be aflat Riemannian space. Then, for any local Minkowski structure F on M, the identity map I:

{M,F)->(M,h)

is harmonic. Proof.

By (4.9) we have

FGi

= i£^-tfi»v.

(9 45)

-

i,k,l

On the other hand, F is local Minkowskian if and only if (see Definition 1.3.1) 9ij(x,y)=

9a{y).

The conclusion is now immediate from (9.44)-(9.46).

(9.46) •

116

An Introduction

to Finsler

Geometry

Recall that a Finsler manifold (M, F) is said to be of Berwald type if F has vanishing Minkowski curvature, i.e., if Pijk\ = 0 for all i,j,k,X (cf. Definition 5.1.3). Recall that a Finsler manifold (M, F) is said to be of Randers type if F = a + B, where a is a Riemannian metric on M, and 0 = Bidx1 is a 1-form. It is easy to see that a Randers manifold if and only if B^j = 0, where fyy is the covariant derivative of f3 with respect to the parallel with respect to a. In this case, the Randers metric a 4-/3 and Riemannian metric a have same geodesic coefficients (cf. [Bao, Chern and Shen, 2000], 11.311). Combining this with (9.44) we obtain: Proposition 9.5.2 Let (M, a 4-/3) be a Randers manifold. If (3 is parallel with respect to the Riemannian metric a, then the identity I:{M,a

+

8)->(M,a)

is harmonic. Antonelli, Ingarden, and Matsumoto (cf. [AIM, 1993]) showed that Berwald nor Minkowskian can be constructed using certain Randers metrics. In view of this, our results gives examples of harmonic maps from Berwald manifolds which are neither Riemannian nor Minkowskian.

Bibliography

T.Aikou, Some remarks on the geometry of tangent bundles of Finsler manifolds, Tensor N. S., 52(1993), 234-241. P.L.Antonelli, R.S.Ingarden, and M.Matsumoto, The theory of sprays and Finsler spaces with appications in physics and biology, Kluwer Academic Publishers, Dordrecht, 1993. H.Akbar-Zadeh, Sur les espaces de Finsler a courbres sectionnelles constantes, Bull. Acad. Roy. Bel. Bull. CI. Sci. (5) 74(1988), 281-322. D.Bao and S.S.Chern, A note on the Gauss-Bonnet Theorem for Finsler Spaces, Ann Math. 143(1996), 233 ~ 252. D. Bao and S. S. Chern, On a notable connection in Finsler geometry, Houston J. Math. 19(1993),135-180. D.Bao, S.S.Chern and Z.Shen, A introduction to Riemannian-Finsler geometry, Graduate Texts in Math, vol. 200, Spinger-Verlag, New York, 2000. D.Bao and C.Robles, On Randers metrics of constant curvature, Reports on Math. Phys.51(2003),9-42. D.Bao, C.Robles and Z.Shen, Zermelo navigatiob on Riemannian spaces, J.Diff.Geom. 66(2004),391-449. D. Bao and Z. Shen, Finsler metrics of constant positive curvature on the Lie group S 3 , J. London Math. Soc. 66(2002), 453-467. D.Bao and Z.Shen, on the Volume of Unit Tangent Spheres is a Finsler Manifold. Results, in. Math. 26(1994), 1-17. X.Chen, X.Mo and Z.Shen, On the flag curvature of Finsler metrics of scalar curvature, J.London Math. Soc. (2)68(2003),762-780. X.Chen and Z.Shen, Randers metrics with special curvature properties, Osaka J.Math. 40(2003), 87-101. S.S. Chern, Local equivalence and Euclidean connection in Finsler space, Sci. Rep. Nat. Tsing Hua Univ. Ser. A5(1948), 95-121. S.S. Chern, On the Finsler geometry, C. R. Acad. Sc. Paris, 314(1992), 757-762. A. Deicke, fiber die Finsler-Raume mit At = 0, Arch. Math. 4(1953), 45-51. J.Eells and L.Lemaire, Select topics of harmonic maps, CBMS Regional Conf. Ser. in Math, vol. 50, Amer. Math. Soc, Providence, RI, 1983. M.Matsumoto, On C-reducible Finsler sapces, Tensor, N.S.24(1972), 29-37. 117

118

Bibliography

M.Matsumoto, Randers spaces of constant curvature, Reports on Math. Phy. 28(1989), 249-261. M.Matsumoto and H.Shimada, The corrected fundamental theorem on Randers spaces of curvature, Tensor, N.S.63(2002), 43-47. M.Matsumoto and S.Hojo, A conclusive theorem for C-reducible Finsler spaces, Tensor. N.S 32(1978), 225-230. X. Mo, Characterization and structure of Finsler spaces with constant flag curvature. Scientia Sinica, 41(1998), 910-917. X. Mo, Flag curvature tensor on a closed Finsler surface. Result in Math., 36(1999) , 149-159. X. Mo, Harmonic maps from Finsler manifolds, Illi. J. Math. 45(2001), 1331-1345. X. Mo, New characterizations of Riemannian spaces, Houston Journal of Mathematics 26(2000) 517-526. Xiaohuan Mo, On the flag curvature of a Finsler space with constant S-curvature, Houston Journal of Mathematics 31(2005),131-144. X.Mo and Z.Shen, On negatively curved Finsler manifolds of scalar curvature, Canada Math. Bull. 48(2005), 112-120. X. Mo and Y. Yang, The existence of harmonic maps from Finsler manifolds to Riemannian manifolds, Scientia Sinica 48(2005), 115-130. S.Numata, On Landsberg spaces of scalar curvature, J.Korea Math. Soc. 12(1975), 97-100. G.Randers, On an asymmetric in the four-space of general relatively, Phys. Rev. 59(1941), 195-199. Z.Shen, Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, Dordrecht(2001), 258 pages. Z.Shen, Finsler metrics with K = 0 and S = 0, Canad. J. Math. 55(2003), 112132. Z.Shen, Lecture on Finsler Geometry, World Scientific Publishers (2001). Z.Shen, Nonpositively curved Finsler manifolds with constant S-curvature, preprint, 2003. Z.Shen, On Finsler geometry of submanifolds, Math. Ann. 311(1998), 549-576. Z.Shen, On some special spaces in Finsler geometry, (1994), Preprint. Z.Shen, Volume comparison and its applications in Riemann-Finsler Geometry, Advances in Math. 128(1997), 306-328. Y.Shen and Y.Zhang, The second variation formula of harmonic maps of Finsler manfolds, Science in China 33(2003), 610-620. Z.Szabo, Posiyive definite Berwald spaces (struture theorems on Berwald spaces), Tensor, N.S. 35(1981)25-39. J.C.Wood, Harmonic morphisms, foliations and Gauss maps, Complex differential geometry and nonlinear differential equations (Brunswick, Maine, 1984), Comtemp. Math, vol. 49, Amer. Math. Soc, Providence, RI, 1986, pp. 145183. H.Yasuda and H.Shimada, On Randers sapces of scalar curvature. Rep. on Math Phys. 11(1977), 347-360.

Index

S-curvature, 44 a-th principal curvature, 58 (globally) Minkowski manifold, 5

coefficients, 35 harmonic, 102 Hilbert form, 9 horizontal subbundle, 37

adapted, 24 Berwald manifold, 57

index form, 68 isotropic S-curvature, 45

Cart an form, 12 tensor, 11 constant S'-curvature, 45 constant (flag) curvature, 59 covariant derivative, 65 curvature 2-form of Chern connection, 53

Klein metric, 4 Landsberg curvature, 42 manifold, 42 locally Minkowski manifold, 5 Matsumoto torsion, 92 mean Berwald curvature, 94 mean curvature, 42, 78 mean Landsberg curvature, 43 minimal, 42 Minkowskian curvature, 54

distinguished section, 24 distortion, 14 dual Finsler bundle, 9 Finsler bundle, 9, 75 manifold, 3 space, 3 structure, 3, 15 flag curvature, 58 flag curvaure tensor, 58 fundamental tensor, 7 Punk metrics, 6

Randers metric, 5 reversible, 9 Ricci scalar, 58 Riemannian, 77 Riemannian curvature, 54

geodesic, 35, 41

scalar curvature, 58

normal curvature, 78

119

120

Index

second fundamental form, 42, 78

vertical subbundle, 37

tension field, 105 totally geodesic, 42, 110

weak Landsberg type, 43 weakly Berwaldian, 94

Peking University Series in Mathematics — Vol. 1

AN INTRODUCTION TO

FINSLER GEOMETRY This i n t r o d u c t o r y book uses the moving frame as a tool and develops Finsler geometry on the basis of the Chern connection and the projective sphere bundle. It systematically introduces t h r e e classes of g e o m e t r i c a l invariants o n Finsler manifolds and their intrinsic relations, analyzes local and global results f r o m classic and m o d e r n Finsler geometry, and gives non-trivial examples of Finsler m a n i f o l d s satisfying different curvature conditions.

6095 he ISBN 981-256-793-3

orld Scientific YEARS OF

PUBLISHING

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Introduction to Finsler geometry

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An introduction to Riemann-Finsler geometry

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Introduction to Algebraic Geometry

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Introduction to algebraic geometry

Introduction to algebraic geometry

Introduction to Algebraic Geometry

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Introduction to Integral Geometry

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A Sampler of Riemann-Finsler Geometry

An Introduction to Algebraical Geometry

Introduction to Computing with Geometry

An Introduction to Riemannian Geometry

An Introduction to Riemannian Geometry

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An Introduction to Riemann-Finsler Geometry

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Lectures on Finsler geometry

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Introduction to Algebraic Geometry

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Introduction to Integral Geometry

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