A counterexample to Matsumoto's conjecture regarding absolute length vs. relative length in Finsler manifolds

TL;DR

This paper disproves Matsumoto's conjecture by providing a counterexample showing that the absolute length of a tangent vector is not always the minimum among relative lengths on the indicatrix in Finsler manifolds.

Contribution

The paper presents the first known counterexample to Matsumoto's conjecture, challenging a previously held assumption in Finsler geometry.

Findings

Counterexample invalidates Matsumoto's conjecture

Absolute length is not always the minimum among relative lengths

Challenges existing beliefs in Finsler geometry

Abstract

Matsumoto conjectured that for any Finsler manifold $(M, F)$ for which the restriction of the fundamental tensor to the indicatrix of $F$ is positive definite, the absolute length $F(X)$ of any tangent vector $X \in T_xM$ is the global minimum for the relative length $|X|_y$ as $y$ varies along the indicatrix $I_x \subset T_xM$ of $F$. In this note, we disprove this conjecture by presenting a counterexample.