On Cartan Spaces with the $m$-th Root Metric $K(x,p)=\sqrt[m]{a^{i_{1}i_{2}...i_{m}}(x)p_{i_{1}}p_{i_{2}}...p_{i_{m}}}$

TL;DR

This paper explores the geometric properties of m-th root Cartan spaces, focusing on the Berwald-Moor metric, including derivations, curvature tensors, and specific conditions for these spaces.

Contribution

It provides a detailed analysis of the geometrical structures of m-th root Cartan spaces with the Berwald-Moor metric, including derivations and curvature tensors, which was not previously studied in detail.

Findings

Computed the v-curvature d-tensor S^{hijk} and analyzed S3-likeness conditions.

Derived the T-tensor T^{hijk} for the m-th root Cartan space.

Specialized results to the case of the Berwald-Moor metric of momenta.

Abstract

The aim of this paper is to expose some geometrical properties of the locally Minkowski-Cartan space with the Berwald-Moor metric of momenta. This space is regarded as a particular case of the $m$-th root Cartan space. Thus, Section 2 studies the $v$-covariant derivation components of the $m$-th root Cartan space. Section 3 computes the $v$-curvature d-tensor $S^{hijk}$ of the m-th root Cartan space and studies conditions for $S3$-likeness. Section 4 computes the $T$-tensor $T^{hijk}$ of the m-th root Cartan space. Section 5 particularizes the preceding geometrical results for the Berwald-Moor metric of momenta.