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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
¡¡1.2 a. euler's theorem !,
¡¡ 1.2 b. a fundamental inequality
¡¡ 1.2 c. interpretations of the fundamental inequality
¡¡¡¡1.3 explicit examples of finsler manifolds
¡¡1.3 a. minkowski and locally minkowski spaces
¡¡ 1.3 b. riemannian manifolds
¡¡ 1.3 c. randers spaces
¡¡1.3 d. berwald spaces
¡¡ 1.3 e. finsler spaces of constant flag curvature
¡¡¡¡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 theorems
¡¡chapter 5 variations of arc length,
¡¡chapter 6 the gauss lemma and the hopf-rinow theorem
¡¡chapter 7 the index form and the bonnet-myers theorem
chapter 8 the cut and conjugate loci, and synge's theorem
chapter 9 the cartan-hadamard theorem and rauch's first theorem
part three special finsler spaces over the reals
chapter 10 berwald spaces and szab6's theorem for berwald surfaces
chapter 11 randers spaces and an elegant theorem
chapter 12 constant flag curvature spaces and akbar-zadeh's theorem
chapter 13 riemannian manifolds and two of hopf's theorems
chapter 14 minkowski spaces, the theorems of deicke and brickell bibliography
index