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

TL;DR

This work develops a projective-invariant Weyl framework for Finsler geometry, introducing the weakly-Weyl and generalized weakly-Weyl metric classes and clarifying their connections to Weyl, Douglas, and $GDW$ invariants through precise curvature identities. It proves the projective invariance of the weakly-Weyl curvature and shows that the generalized class is closed under projective changes, enabling streamlined proofs of Sakaguchi-type results. In particular, it establishes that every weakly-Weyl Finsler metric is a $GDW$-metric and characterizes weakly-Weyl spherical metrics in $\mathbb{R}^n$ as exactly the $W$-quadratic ones, with explicit Weyl-curvature formulas. The generalized weakly-Weyl framework is shown to intersect with but not be contained in the $GDW$ class, suggesting a unified yet diverse landscape of projective invariants and motivating further exploration of non-$GDW$ examples.

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.

Theorems & Definitions (22)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Lemma 3.1
• Definition 4.1
• Proposition 4.2
• proof
• Definition 4.3