內容介紹
This book project began as an attempt to sort through the literature on Finsler geometry. It was our intention to write a systematic account about that part of the material which is both elementary and indispensable. We want to thank many fellow geometers for their encouragement, for answering our email calls for help, and for steering us towards the pertinent references. Some of these colleagues also helped us by proof-reading parts of the manuscript. ...作品目錄
preface.acknowledgments
part one finsler manifolds and their curvature
chapter 1 finsler manifolds and the fundamentals of minkowski norms
1.0 physical motivations
1.1 finsler structures: definitions and conventions
1.2 two basic properties of minkowski norms
1.3 explicit examples of finsler manifolds
1.4 the fundamental tensor and the cartan tensor
references for chapter 1
chapter 2 the chern connection
2.0 prologue
2.1 the vector bundle tm and related objects
2.2 coordinate bases versus special orthonormal bases
2.3 the nonlinear connection on the manifold tm
2.4 the chern connection on tm
2.5 index gymnastics
references for chapter 2
chapter 3 curvature and schur's lemma
3.1 conventions and the hh-, hv-, w-curvatures
.3.2 first bianchi identities from torsion freeness
3.3 formulas for r and p in natural coordinates
3.4 first bianchi identities from "almost" g-compatibility
3.5 second bianchi identities
3.6 interchange formulas or ricci identities
3.7 lie brackets among the and the
3.8 derivatives of the geodesic spray coefficients gi
3.9 the flag curvature
3.10 schur's lemma
references for chapter 3
chapter 4 finsler surfaces and a generalized gauss-bonnet theorem
4.0 prologue
4.1 minkowski planes and a useful basis
4.2 the equivalence problem for minkowski planes
4.3 the berwald frame and our geometrical setup on sm
4.4 the chern connection and the invariants i, j, k
4.5 the riemannian arc length of the indicatrix
4.6 a gauss-bonnet theorem for landsberg surfaces
references for chapter 4
part two calculus of variations and comparison theoremschapter 5
variations of arc length,
jacobi fields, the effect of curvature
5.1 the first variation of arc length
5.2 the second variation of arc length
5.3 geodesics and the exponential map
5.4 jacobi fields
5.5 how the flag curvature's sign influences geodesic rays
references for chapter 5
chapter 6
the gauss lemma and the hopf-rinow theorem
6.1 the gauss lemma
6.2 finsler manifolds and metric spaces
6.3 short geodesics are minimizing
6.4 the smoothness of distance functions
6.5 long minimizing geodesics
6.6 the hopf-rinow theorem
chapter 7
the index form and the bonnet-myers theorem
7.1 conjugate points
7.2 the index form
7.3 what happens in the absence of conjugate points?
7.4 what happens if conjugate points are present?
7.5 the cut point versus the first conjugate points
7.6 ricci curvatures
7.7 the bonnet-myers theorem
references for chapter 7..
chapter 8
the cut and conjugate loci, and synge's theorem
8.1 definitions
8.2 the cut point and the first conjugate point
8.3 some consequences of the inverse function theorem
8.4 the manner in which cy and iy depend on y
8.5 generic properties of the cut locus cutx
8.6 additional properties of cuts when m is compact
8.7 shortest geodesics within homotopy classes
8.8 synge's theorem
references for chapter 8
chapter 9
the cartan-hadamard theorem and
rauch's first theorem
9.1 estimating the growth of jacobi fields
9.2 when do local diffeomorphisms become covering maps?
9.3 some consequences of the covering homotopy theorem
9.4 the cartan-hadamard theorem
9.5 prelude to ranch's theorem
9.6 rauch's first comparison theorem
9.7 jacobi fields on space forms
9.8 applications of rauch's theorem
references for chapter 9
part three
special finsler spaces over the reals
chapter 10
berwald spaces and
szabo's theorem for berwald surfaces
10.0 prologue
10.1 berwald spaces
10.2 various characterizations of berwald spaces
10.3 examples of berwald spaces
10.4 a fact about flat linear connections
10.5 characterizing locally minkowski spaces by curvature
10.6 szabo's rigidity theorem for berwald surfaces
references for chapter 10
chapter 11
randers spaces and an elegant theorem
11.0 the importance of randers spaces
11.1 randers spaces, positivity, and strong convexity
11.2 a matrix result and its consequences
11.3 the geodesic spray coefficients of a randers metric
11.4 the nonlinear connection for randers spaces
11.5 a useful and elegant theorem
11.6 the construction of y-global berwald spaces
references for chapter 11
chapter 12
constant flag curvature spaces and
akbar-zadeh's theorem
12.0 prologue
12.1 characterizations of constant flag curvature
12.2 useful interpretations of e and e
12.3 growth rates of solutions of e + λe = 0
12.4 akbar-zadeh's rigidity theorem
12.5 formulas for machine computations of k
12.6 a poincare disc that is only forward complete
12.7 non-riemannian projectively flat s2 with k = 1
references for chapter 12
chapter 13
riemannian manifolds and two of hopf's theorems
13.1 the levi-civita (christoffel) connection
13.2 curvature
13.3 warped products and riemannian space forms
13.4 hopf's classification of riemannian space forms
13.5 the divergence lemma and hopf's theorem.
13.6 the weitzenbsck formula and the bochner technique
references for chapter 13
chapter 14
minkowski spaces, the theorems of deicke and brickell
14.1 generalities and examples
14.2 the riemannian curvature of each minkowski space
14.3 the riemannian laplacian in spherical coordinates
14.4 deicke's theorem
14.5 the extrinsic curvature of the level spheres of f
14.6 the gauss equations
14.7 the blaschke-santal6 inequality
14.8 the legendre transformation
14.9 a mixed-volume inequality, and brickell's theorem
references for chapter 14
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