Generalized Weakly-Weyl Finsler Metrics: A Generalized Approach to Sakaguchi's Theorem

TL;DR

This paper introduces a new class of projective invariant Finsler metrics, explores their properties, and establishes their connection to scalar flag curvature, expanding the understanding of generalized Sakaguchi's Theorem.

Contribution

It defines generalized weakly-Weyl Finsler metrics, links them to projective invariants, and proves their equivalence with W-quadratic spherically symmetric metrics in Euclidean space.

Findings

Weakly-Weyl and W-quadratic spherically symmetric metrics are equivalent in ℝ^n.

Every Finsler metric of scalar flag curvature is a GDW-metric.

New examples of projective invariant Finsler metrics are provided.

Abstract

The development of projective invariant Weyl metrics in this paper offers a fresh perspective, as we establish the characteristics of both weakly-Weyl and generalized weakly-Weyl Finsler metrics. We thoroughly examine the connections between these metrics and various projective invariants, highlighting their significance in the context of generalized Sakaguchi's Theorem, which states that every Finsler metric of scalar flag curvature is a GDW-metric. Additionally, we introduce several illustrative examples pertaining to this new class of projective invariant Finsler metrics. Specifically, we explore the category of weakly-Weyl spherically symmetric Finsler metrics in $\mathbb{R}^n$. Importantly, we demonstrate that the two classes weakly-Weyl and $W$-quadratic spherically symmetric Finsler metrics in $\mathbb{R}^n$ are equivalent.