Preface

Foundations

1.  Differentiable Manifolds

  1.1  Differentiable manifolds

    1.1.1  Differentiable manifolds

    1.1.2  Examples of differentiable manifolds

  1.2  Vector fields and tensor fields

    1.2.1  Vector bundles

    1.2.2  Tensor fields

  1.3  Exterior forms and exterior differentials

    1.3.1  Exterior differential operators

    1.3.2  de Rham theorem

  1.4  Vector bundles and connections

    1.4.1  Connection of the vector bundle

    1.4.2  Curvature of a connection

  Exercises

2.  Finsler Metrics

  2.1  Finsler metrics

    2.1.1  Finsler metrics

    2.1.2  Examples of Finsler metrics

  2.2  Cartan torsion

    2.2.1  Cartan torsion

    2.2.2  Deicke theorem

  2.3  Hilbert form and sprays

    2.3.1  Hilbert form

    2.3.2  Sprays

  2.4  Geodesics

    2.4.1  Geodesics

    2.4.2  Geodesic coefficients

    2.4.3  Geodesic completeness

  Exercises

3.  Connections and Curvatures

  3.1  Connections

    3.1.1  Chern connection

    3.1.2  Berwald metrics and Landsberg metrics

  3.2  Curvatures

    3.2.1  Curvature form of the Chern connection

    3.2.2  Flag curvature and Ricci curvature

  3.3  Bianchi identities

    3.3.1  Covariant differentiation

    3.3.2  Bianchi identities

    3.3.3  Other formulas

  3.4  Legendre transformation

    3.4.1  The dual norm in the dual space

    3.4.2  Legendre transformation

    3.4.3  Example

  Exercises

4.  S-Curvature

  4.1  Volume measures

    4.1.1  Busemann-Hausdorff volume element

    4.1.2  The volume element induced from SM

  4.2  S-curvature

    4.2.1  Distortion

    4.2.2  S-curvature and E-curvature

  4.3  Isotropic S-curvature

    4.3.1  Isotropic S-curvature and isotropic E-curvature

    4.3.2  Randers metrics of isotropic S-curvature

    4.3.3  Geodesic flow

  Exercises

5.  Riemann Curvature

  5.1  The second variation of arc length

    5.1.1  The second variation of length

    5.1.2  Elements of curvature and topology

  5.2  Scalar flag curvature

    5.2.1  Schur theorem

    5.2.2  Constant flag curvature

  5.3  Global rigidity results

    5.3.1  Flag curvature with special conditions

    5.3.2  Manifolds with non-positive flag curvature

  5.4  Navigation

    5.4.1  Navigation problem

    5.4.2  Randers metrics and navigation

    5.4.3  Ricci curvature and Einstein metrics

  Exercises

  Further Studies

6.  Projective Changes

  6.1  The projective equivalence

    6.1.1  Projective equivalence

    6.1.2  Projective invariants

  6.2  Projectively flat metrics

    6.2.1  Projectively flat metrics

    6.2.2  Projectively fiat metrics with constant flag curvature

  6.3  Projectively fiat metrics with almost isotropic S-curvature

    6.3.1  Randers metrics with almost isotropic S-curvature

    6.3.2  Projectively flat metrics with almost isotropic

  S-curvature

  6.4  Some special projectively equivalent Finsler metrics

    6.4.1  Projectively equivalent Randers metrics

    6.4.2  The projective equivalence of (α, β)-metrics

    6.4.3  The projective equivalence of quadratic (α, β)

  metrics

  Exercises

7.  Comparison Theorems

  7.1  Volume comparison theorems for Finsler manifolds

    7.1.1  The Jacobian of the exponential map

    7.1.2  Distance function and comparison theorems

    7.1.3  Volume comparison theorems

  7.2  Berger-Kazdan comparison theorems

    7.2.1  The Kazdan inequality

    7.2.2  The rigidity of reversible Finsler manifolds

    7.2.3  The Berger-Kazdan comparison theorem

  Exercises

8.  Fundamental Groups of Finsler Manifolds

  8.1  Fundamental groups of Finsler manifolds

    8.1.1  Fundamental groups and covering spaces

    8.1.2  Algebraic norms and geometric norms

    8.1.3  Growth of fundamental groups

  8.2  Entropy and finiteness of fundamental group

    8.2.1  Entropy of fundamental group

    8.2.2  The first Betti number

    8.2.3  Finiteness of fundamental group

  8.3  Gromov pre-compactness theorems

    8.3.1  General metric sp