Introduction to modern Finsler geometry

Description

Intro Contents 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 torsion2.2.2 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 transformation3.4.3 Example Exercises 4. S-Curvature 4.1 Volume measures 4.1.1 Busemann-Hausdor 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 curvature5.3 Global rigidity results 5.3.1 Flag curvature with special conditions 5.3.2 Manifolds with non-positive ag 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 at metrics 6.2.1 Projectively at metrics 6.2.2 Projectively at metrics with constant ag curvature 6.3 Projectively at metrics with almost isotropic S-curvature. 6.3.1 Randers metrics with almost isotropic S-curvature6.3.2 Projectively at 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 ()-metrics 6.4.3 The projective equivalence of quadratic ()-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.