Major Topics in Finsler Geometry

 

1. Introduction to Finsler Geometry

1.1 What is Finsler Geometry?
1.2 Historical Development and Motivation
1.3 Comparison: Euclidean, Riemannian, and Finsler Metrics
1.4 Applications in Modern Geometry, Physics, and Data Science
1.5 Notation and Conventions

2. Finsler Structures on Manifolds

2.1 The Tangent Bundle TMTM and Coordinates
2.2 The Finsler Function F:TM→[0,∞)F: TM \rightarrow [0, \infty)
2.3 Homogeneity and Regularity Conditions
2.4 The Energy Function E=12F2E = \frac{1}{2}F^2
2.5 Examples of Finsler Structures (Randers, Kropina, Matsumoto)

3. Fundamental Tensors and Metric Structures

3.1 The Fundamental Metric Tensor gij(x,y)g_{ij}(x, y)
3.2 The Cartan Tensor Cijk=12∂gij/∂ykC_{ijk} = \frac{1}{2} \partial g_{ij} / \partial y^k
3.3 Angular Metric hijh_{ij} and Its Role
3.4 Projective and Conformal Transformations in Finsler Geometry
3.5 The Role of Minkowski Norms in Tangent Spaces

4. Nonlinear Connections

4.1 Definition and Motivation
4.2 Horizontal and Vertical Distributions
4.3 Berwald’s Nonlinear Connection
4.4 Chern’s Nonlinear Connection
4.5 Splitting of the Tangent Bundle TTMTTM: Ehresmann Connections

5. Finsler Connections and Covariant Derivatives

5.1 The Cartan Connection
5.2 The Berwald Connection
5.3 The Chern (Rund) Connection
5.4 The Hashiguchi and Matsumoto Connections
5.5 Torsion and Curvature of Finsler Connections
5.6 Comparison and Properties of Different Finsler Connections

6. Geodesics and Variational Structure

6.1 Finsler Geodesics via Euler–Lagrange Equations
6.2 Canonical Spray and Geodesic Flow
6.3 Projective Equivalence of Geodesics
6.4 Jacobi Fields and Stability of Geodesics
6.5 Completeness, Hopf–Rinow Theorems, and Distance Functions

7. Curvature in Finsler Geometry

7.1 Riemann Curvature: Horizontal (h-) Curvature
7.2 Berwald Curvature and Its Geometric Meaning
7.3 Landsberg Curvature and Mean Landsberg Tensor
7.4 Flag Curvature: Finslerian Generalization of Sectional Curvature
7.5 Ricci and Scalar Curvatures in the Finsler Context
7.6 ฯ‡-Curvature (Chi Curvature) and Finsler Integrability

8. Special Finsler Spaces

8.1 Riemannian and Locally Minkowskian Spaces
8.2 Berwald and Weakly Berwald Spaces
8.3 Landsberg and Weak Landsberg Spaces
8.4 Kropina and Randers Spaces
8.5 Isotropic and Constant Flag Curvature Spaces

9. Symmetries and Conservation Laws

9.1 Killing Vector Fields in Finsler Geometry
9.2 Conformal and Projective Symmetries
9.3 Killing Tensors and Generalizations
9.4 First Integrals from Symmetries (Noether-type Theorems)
9.5 Sarlet's Recursive Scheme and Higher-Order Integrals

10. Comparison Geometry and Topology

10.1 Comparison Theorems for Flag Curvature
10.2 Sphere Theorems in Finsler Geometry
10.3 Volume Comparison and the Busemann–Hausdorff Measure
10.4 Topological Rigidity and Curvature Bounds
10.5 Finslerian Generalizations of Bonnet–Myers

11. Applications in Physics and Geometry

11.1 Lorentz–Finsler Geometry and Spacetime Models
11.2 Anisotropic Relativity and Modified Dispersion Relations
11.3 Finslerian Optics and Wave Propagation
11.4 Finsler Structures in Biological and Cognitive Models
11.5 Finsler Metrics in Data Analysis and Machine Learning

12. Computational Aspects

12.1 Numerical Integration of Geodesics
12.2 Computation of Fundamental and Cartan Tensors
12.3 Discrete Finsler Metrics and Polyhedral Approximations
12.4 Symbolic Computation in Finsler Geometry
12.5 Software Tools (SageMath, Maple, TensorFlow Geometry)

13. Advanced Topics and Open Problems

13.1 Global Problems in Finsler Geometry
13.2 Finslerian Ricci Flow and Metric Evolution
13.3 Spectral Geometry and Eigenvalue Problems
13.4 Connections to Topos Theory and Homotopy Type Theory
13.5 Finsler Structures in Non-Commutative and Quantum Geometry

Appendices

A. Key Definitions and Notation
B. Coordinate-Free Formulations
C. Comparison Tables of Connections
D. Open Problem List in Finsler Research
E. Bibliographic Guide and Historical Timeline

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