芬斯勒几何中的比较定理与子流形（英文版）

Chapter 1Basics on Finsler Geometry1
1。1Minkowski Space1
1。1。1Definition and Examples1
1。1。2Legendre Transformation6
1。1。3Cartan Tensor10
1。2Finsler Manifold12
1。2。1The Definition of Finsler Manifold12
1。2。2Connection and Curvature 13
1。3Geodesic16
1。3。1Geodesic and Exponential Map16
1。3。2The First Variation ofArc Length18
1。3。3The Second Variation ofArc Length25
1。4Jacobi Fields and ConjugatePoints28
1。4。1Jacobi Fields28
1。4。2Conjugate Points32
1。5Basic Index Lemma33
Chapter 2Comparison Theorems in Finsler Geometry39
2。1Rauch Comparison Theorem39
2。2VolumeForm43
2。2。1Definition and Examples43
2。2。2Distortion and S-Curvature 47
2。3Hessian Comparison Theorem and Laplacian Comparison Theorem48
2。3。1Polar Coordinates 48
2。3。2Hessian Comparison Theorem51
2。3。3Laplacian Comparison Theorem53
2。4VolumeComparison Theorems(I): Pointwise Curvature Bounds56
2。5VolumeComparison Theorems(II): Integral Curvature Bounds62
2。6VolumeComparison Theorems(III): Tubular Neighborhoods72
2。6。1Fermi Coordinates for Minkowski Space73
2。6。2Jacobi Fields with Initial Submanifolds75
2。6。3Fermi Coordinates and Focal Cut Locus80
2。6。4Volume Comparison Theorem for Tubular Neighborhoods of Submanifolds82
2。7Comparison Theorems with Weighted Curvature Bounds85
2。8Toponogov Type Comparison Theorem94
Chapter 3Applications of Comparison Theorems98
3。1Generalized Myers Theorem and Linearly Growth Theorem of Volume98
3。1。1Generalized Myers Theorem98
3。1。2Linearly Growth Theorem of Volume100
3。2McKean Type Inequalities for the First Eigenvalue101
3。2。1The Divergence Lemma101
3。2。2The Mckean Type Inequalities103
3。3Gromov Pre-Compactness Theorem107
3。4The First Betti Number109
3。5Curvature and Fundamental Group114
3。5。1Universal Covering Space and Fundamental Group114
3。5。2Growth of Fundamental Group118
3。5。3Finiteness of Fundamental Group124
3。5。4Results Related to Milnor’s Conjecture125
3。6A Lower Bound of Injectivity Radius128
3。7Finite Topological Type131
Chapter 4Geometry of Finsler Submanifolds135
4。1Mean Curvature135
4。1。1Projection in a Minkowski Space135
4。1。2The Mean Curvature for Finsler Submanifolds137
4。2Some Results on Submanifolds in Minkowski Space139
4。3Volume Growth of Submanifolds in Minkowski Space145
4。4Rigidity of Minimal Surfaces in Randers-Minkowski 3-Space149
4。4。1The Mean Curvature of a Graph in (Rn+1，Fb)149
4。4。2The Rigidity Results153
Bibliography156
Index160

Chapter 1 Basics on Finsler Geometry
Finsler geometry is just the Riemannian geometry without the quadratic restriction。 Instead of an Euclidean norm on each tangent space， one endows every tangent space of a differentiable manifold with a Minkowski norm。 Since there are many different Minkowski spaces that are not mutually isomorphic， Finsler manifolds are more “colorful” than Riemannian manifolds。 In this chapter we shall give a brief description of basic quantities and fundamental properties for Finsler metrics which axe foundations to study global Finsler geometry。
1。1Minkowski Space
1。1。1Definition and Examples
Before giving the definition of Minkowski space， let us first discuss a useful property of positively homogeneous function on n-dimensional space of real number Rn。 Re?call that a function f : Mn R is called a positively homogeneous function of degree s on Rn if f(Xy) = Xs f (y) holds for any y G Rn and A > 0。
Lemma 1。1 (Euler’s Lemma) Let f : Mn R be a positively homogeneous function of degree s on then fy% (y)yl = sf(y)， here we have used the the Ein-stein convention，that is，repeated indices with one upper index and one lower index denotes summation over their ranges， and fyi denotes the partial derivative of f with respect to yl。
Proof Taking derivative on two sides of /(Ay) = Xsf (y) with respect to A yields fyi (Xy)yl = sXs~1f(y)。 Letting A = 1 we get the desired result。□
Now we give the definition of Minkowski space。
Definition 1。1 Let V be an n-dimensional real vector space。 A function F = F(y) on V is called a Minkowski norm if it satisfies the following properties:
(1)F(y)>= 0 for any y and F(y) = 0 if and only if y = 0；
(2)F(Xy) = XF(y) for any y EV and A > 0；
(3)F is C°° on ^{0} such that for any y E F{0}， the following bilinear symmetric functional on V is an inner product:
The pair (V， F) is called a Minkowski space， and gy the fundamental form with respect to y ， If F(y) = F(-y) holds for any y G V， then (V， F) is called reversible。 Let (V， F) be a Minkowski space， and
S = {y& VF(y) = 1}。
5 is a closed hypersurface around the origin， which is diffeomorphic to the standard sphere Sn_1 C Mn。 S is called the indicatrix of (V， F)。 Fix a basis {b^} of V， and view F(y) = F(ylhi) as a function of (yl) G Mn。 For y 0，write
9ij(y) =^[F2]yiyj (y) = Fyi (y)Fyj (y) + F(y)Fyiyj (y)。
Then
gy(u，v) = gij(y)ulvj， u = ulbi，v = vlbi。
Since the Minkowski norm is a positively homogeneous function of degree Euler’s lemma we get
When (V， F) is reversible， it is clear that gij(y) = gij(-y)， and consequently gy(u， v)
For any y ^0， one has the following decomposition for V:
V = R-yQWy，
here Wy is the orthogonal complementary of R y in V with respect to the inner product gy， namely，
Let1
We call hy the angular form with respect to y， and its components are
It is easy to know that hy(y， u) = 0， Vw G V， and fov u = wXy ^ V with w G Wy， one has and the equality holds if and only if u = Xy。 Hence， is semi-positive definite on V while it is positive definite on Wy。 Now we prove two fundamental inequalities for Minkowski space。
Lemma 1。2 Let (V， F) be a Minkowski space，then
(1)F satisfies
and the equality holds if and only if u = 0 or there zs A ^ 0 such that v = An；
(2)(Cauchy-Schwarz Inequality) Let y- Q， then for any u EV one has
and the equality holds if and only if there exists A ^ 0 such that u = Xy。
Proof Let us first prove (1。3)。 Without loss of generality， we can assume that u，v E y{0}。 If u，v are linearly independent， let y(t) = tu(1 – t)v， then y(t) 0， t G [0，1]。 Consider the function
Therefore， (p = (p(t) is a strict convex function， and we see from the property of convex function that namely，
When u，v are linearly dependent， then there exists A G M such that v = Xu。 If 1 + λ>= 0，then
F(u + v) = F((l + X)u) = (1 + A)F(u) < F(u) + F(Xu) = F(u) + F(v)， and the equality holds if and only if A ^ 0。 Finally， when 1 + A < 0，
F(u + v) = F(-(1 + X)(-u)) = -(1 + X)F(-u) = -F(-u) + F(Xu) < F(u) + F(v)，
and thus (1。3) is proved。
Now let us prove (1。4)。 For w G Wy， let (p(t) = F2(y + tw)， then by (1。1)，(1。2) and Euler’s lemma we have and the equality holds if and only if = 0。 Consequently， ^(0)(p(i)^t + 0， and the equality holds if and only if w = 0。 Especially，
with the equality holds if and only if 忉=0。 Now for any u eV， write u = Xy –w， here A G M， ^ G Wy。 Then
If λ≥ 0， then it is easy to see from (1。6) that (1。4) holds， with the equality holds if and only if 入=0 and u = 0。 On the other hand， if A > 0， it is clear from (1。5) and (1。6) that
Sy(y，u) = XF(y)F(y) ^ XF (y + F(y) = F(u)F(y)。
The equality holds if and only if ty = 0， namely， u = Ay， A ≥ 0， so the lemma is proved。
From (1。3) it is clear that F is the norm of vector space V in the usual sense when F is reversible， and it is also easy to verify that (1。4) can be rewritten as
when (V， F) is reversible。 But (1。4)； does not hold for general Minkowski space (see Example 1。2)。
In the following we give some important examples of Min