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Characteristic tensors for almost Finsler manifolds

James F. Davis, Benjamin R. Edwards, Alan Kostelecky

Abstract

Almost Finsler manifolds and partial Finsler manifolds are introduced, extending the standard definition of a Finsler manifold to allow for a nontrivial slit containing points fixed under homogeneous scaling and for metrics where the fundamental tensor has nonpositive eigenvalues. The bipartite spaces offer examples of comparatively simple almost Finsler manifolds and partial Finsler manifolds with physics applications. Special cases are the $\bf{a}$ and $\bf{b}$ spaces, which have almost Finsler norms and partial Finsler norms formed from a Riemannian norm and a 1-form. The indicatrix union of the almost Finsler $\bf{a}$ manifolds equals the indicatrix union of Randers spaces. Characteristic tensors that vanish for bipartite spaces and $\bf{b}$ spaces are obtained and expressed using geometric quantities. These tensors are generalizations of the Matsumoto tensor, which vanishes on Randers and $\bf{a}$ spaces.

Characteristic tensors for almost Finsler manifolds

Abstract

Almost Finsler manifolds and partial Finsler manifolds are introduced, extending the standard definition of a Finsler manifold to allow for a nontrivial slit containing points fixed under homogeneous scaling and for metrics where the fundamental tensor has nonpositive eigenvalues. The bipartite spaces offer examples of comparatively simple almost Finsler manifolds and partial Finsler manifolds with physics applications. Special cases are the and spaces, which have almost Finsler norms and partial Finsler norms formed from a Riemannian norm and a 1-form. The indicatrix union of the almost Finsler manifolds equals the indicatrix union of Randers spaces. Characteristic tensors that vanish for bipartite spaces and spaces are obtained and expressed using geometric quantities. These tensors are generalizations of the Matsumoto tensor, which vanishes on Randers and spaces.

Paper Structure

This paper contains 14 sections, 20 theorems, 38 equations, 1 figure.

Table of Contents

  1. Introduction and summary
  2. Almost and partial Finsler manifolds
  3. Minkowski norms
  4. Finsler manifolds
  5. Tensors
  6. Tensors and tensor fields
  7. Components and coordinates
  8. Contraction; lowering and raising indices
  9. Bipartite spaces
  10. Basics
  11. Bipartite indicatrix unions
  12. Characteristic tensors
  13. Proof of Theorem \ref{['bipthm']}
  14. Proof of Theorem \ref{['bthm']}

Key Result

Theorem 1

The characteristic tensor vanishes for almost Finsler manifolds that are bipartite spaces.

Figures (1)

  • Figure 1: The bipartite indicatrix union $\widehat{I}_{{\mathbf b}}$ for $n=3$.

Theorems & Definitions (56)

  • Theorem 1
  • Theorem 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Example 6
  • Example 7
  • Example 8
  • Definition 9
  • Definition 10
  • ...and 46 more