Anisotropic tensor calculus

TL;DR

This paper develops a new calculus for anisotropic tensors that preserves tensor class under derivatives, enabling advanced geometric analysis and reinterpretation of connections in Finsler geometry.

Contribution

It introduces anisotropic tensor calculus, defines anisotropic derivations, and relates these to classical connections, including a novel interpretation of the Chern connection.

Findings

Anisotropic tensor calculus maintains tensor class under differentiation.

The Chern connection of a Finsler metric can be viewed as a Levi-Civita connection.

Introduces anisotropic curvature tensor and anisotropic Lie derivative.

Abstract

We introduce the anisotropic tensor calculus, which is a way of handling with tensors that depend on the direction remaining always in the same class. This means that the derivative of an anisotropic tensor is a tensor of the same type. As an application, we show how to define derivations using anisotropic linear connections in a manifold. In particular, we show that the Chern connection of a Finsler metric can be interpreted as the Levi-Civita connection and we introduce the anisotropic curvature tensor. We also relate the concept of anisotropic connection with the classical concept of linear connections in the vertical bundle. Furthermore, we also introduce the concept of anisotropic Lie derivative.