On the Cartan torsion of Minkowskian product of Finsler manifolds

TL;DR

This paper studies the Cartan torsion of Minkowskian products of Finsler manifolds, deriving formulas and conditions for their geometric properties, especially focusing on when torsions decompose additively and remain bounded.

Contribution

It provides explicit formulas for Cartan torsion in Minkowskian products and characterizes when torsions decompose additively and stay bounded.

Findings

Torsions decompose additively iff the product is Euclidean

Explicit formulas for Cartan torsion of Minkowskian products

Bounded mean Cartan torsion occurs in Euclidean cases

Abstract

In this paper, we investigate the Cartan torsion and mean Cartan torsion of the Minkowskian product of two Finsler metrics. We derive explicit formulas for both torsions and analyze their geometric behavior. In particular, we demonstrate that both the Cartan torsion and mean Cartan torsion decompose additively if and only if the Minkowskian product is Euclidean. Additionally, we provide a necessary and sufficient condition for the norm of the mean Cartan torsion to remain bounded in the Euclidean case.