Introduction to Modern Finsler Geometry

The following sections are included:

• Preface
• Contents

The following sections are included:

• Differentiable manifolds
  • Differentiable manifolds
  • Examples of differentiable manifolds
• Vector fields and tensor fields
  • Vector bundles
  • Tensor fields
• Exterior forms and exterior differentials
  • Exterior differential operators
  • de Rham theorem
• Vector bundles and connections
  • Connection of the vector bundle
  • Curvature of a connection
• Exercises

The following sections are included:

• Finsler metrics
  • Finsler metrics
  • Examples of Finsler metrics
• Cartan torsion
  • Cartan torsion
  • Deicke theorem
• Hilbert form and sprays
  • Hilbert form
  • Sprays
• Geodesics
  • Geodesics
  • Geodesic coefficients
  • Geodesic completeness
• Exercises

The following sections are included:

• Connections
  • Chern connection
  • Berwald metrics and Landsberg metrics
• Curvatures
  • Curvature form of the Chern connection
  • Flag curvature and Ricci curvature
• Bianchi identities
  • Covariant differentiation
  • Bianchi identities
  • Other formulas
• Legendre transformation
  • The dual norm in the dual space
  • Legendre transformation
  • Example
• Exercises

The following sections are included:

• Volume measures
  • Busemann-Hausdorff volume element
  • The volume element induced from SM
• S-curvature
  • Distortion
  • S-curvature and E-curvature
• Isotropic S-curvature
  • Isotropic S-curvature and isotropic E-curvature
  • Randers metrics of isotropic S-curvature
  • Geodesic flow
• Exercises

The following sections are included:

• The second variation of arc length
  • The second variation of length
  • Elements of curvature and topology
• Scalar flag curvature
  • Schur theorem
  • Constant flag curvature
• Global rigidity results
  • Flag curvature with special conditions
  • Manifolds with non-positive flag curvature
• Navigation
  • Navigation problem
  • Randers metrics and navigation
  • Ricci curvature and Einstein metrics
• Exercises

The following sections are included:

• The projective equivalence
  • Projective equivalence
  • Projective invariants
• Projectively flat metrics
  • Projectively flat metrics
  • Projectively flat metrics with constant flag curvature
• Projectively flat metrics with almost isotropic S-curvature
  • Randers metrics with almost isotropic S-curvature
  • Projectively flat metrics with almost isotropic S-curvature
• Some special projectively equivalent Finsler metrics
  • Projectively equivalent Randers metrics
  • The projective equivalence of (α, β)-metrics
  • The projective equivalence of quadratic (α, β)-metrics
• Exercises

The following sections are included:

• Volume comparison theorems for Finsler manifolds
  • The Jacobian of the exponential map
  • Distance function and comparison theorems
  • Volume comparison theorems
• Berger-Kazdan comparison theorems
  • The Kazdan inequality
  • The rigidity of reversible Finsler manifolds
  • The Berger-Kazdan comparison theorem
• Exercises

The following sections are included:

• Fundamental groups of Finsler manifolds
  • Fundamental groups and covering spaces
  • Algebraic norms and geometric norms
  • Growth of fundamental groups
• Entropy and finiteness of fundamental group
  • Entropy of fundamental group
  • The first Betti number
  • Finiteness of fundamental group
• Gromov pre-compactness theorems
  • General metric spaces
  • δ-Gromov-Hausdor convergence
  • Pre-compactness of Finsler manifolds
  • On the Gauss-Bonnet-Chern theorem
• Exercises

The following sections are included:

• Isometric immersions
  • Finsler submanifolds
  • The variation of the volume
  • Non-existence of compact minimal submanifolds
• Rigidity of minimal submanifolds
  • Minimal surfaces in Minkowski spaces
  • Minimal surfaces in (α, β)-spaces
  • Minimal surfaces in special Minkowskian (α, β)-spaces
• Harmonic maps
  • A divergence formula
  • Harmonic maps
  • Composition maps
• Second variation of harmonic maps
  • The second variation
  • Stress-energy tensor
• Harmonic maps between complex Finsler manifolds
  • Complex Finsler manifolds
  • Harmonic maps between complex Finsler manifolds
  • Holomorphic maps
• Exercises

In this chapter, we will focus on Einstein metrics in Finsler geometry. By Definition 5.4 (§5.4.3), a Finsler metric F = F(x, y) on an n-dimensional manifold M is called an Einstein metric if its Ricci curvature satisfies Ric = (n − 1)K(x)F². Moreover, it is said to have Einstein constant σ if K(x) = σ = constant…

The following sections are included:

• Conformal changes
  • Conformal changes
  • Conformally flat metrics
  • Conformally flat (α, β)-metrics
• Conformal vector fields
  • Conformal vector fields
  • Conformal vector fields on a Randers manifold
• A class of critical Finsler metrics
  • The Einstein-Hilbert functional
  • Some special ε-critical metrics
• The first eigenvalue of Finsler Laplacian and the generalized maximal principle
  • Finsler Laplacian and weighted Ricci curvature
  • Lichnerowicz-Obata estimates
  • Li-Yau-Zhong-Yang type estimates
  • Mckean type estimates
• Exercises

In Finsler geometry, the computations of geometric quantities are usually very complicated. However, using a Maple program, we can quickly check whether or not an expression is correct even though we can derive it manually; we can simplify the expressions in the computation without making mistakes.

The following sections are included:

• Bibliography
• Index