# Characteristic tensors for almost Finsler manifolds

## TL;DR

This paper introduces almost Finsler and partial Finsler manifolds, extending classical Finsler geometry, and defines characteristic tensors that generalize the Matsumoto tensor, with applications in physics.

## Contribution

It extends Finsler geometry to include almost and partial Finsler manifolds, introduces characteristic tensors, and explores their properties and applications.

## Key findings

- Characteristic tensors vanish for bipartite and b-spaces.
- Indicatrix union of almost Finsler a-spaces equals that of Randers spaces.
- Generalization of the Matsumoto tensor to broader classes of 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 $\bf{a}$ 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.