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New classes of Finsler metrics: The birth of new projective invatiant

Nasrin Sadeghzadeh

TL;DR

The paper introduces a new projective invariant in Finsler geometry, the weakly generalized Douglas-Weyl (W-GDW) metrics, defined by the equation tildeD_j^i_{kl|0} + lambda F tildeD_j^i_{kl} = U_{jkl} y^i, and shows this class is closed under projective changes while containing all GDW metrics. It identifies two new subclasses, generalized tilde-D metrics and generalized weakly-Weyl metrics, and proves that every GDW-metric is a W-GDW-metric, while also deriving nontrivial conditions under which a regular (α,β)-metric becomes a W-GDW-metric without being GDW. The work connects these new invariants to existing projective invariants (Weyl, Douglas) and clarifies hierarchical relationships via lemmas, propositions, and examples. Overall, it broadens the landscape of projective invariants in Finsler geometry and provides a unified framework for analyzing curvature-based properties under projective transformations, with implications for αβ-metrics and their curvature theories.

Abstract

This paper presents a pioneering projective invariant in Finsler geometry, introducing a new class of Finsler metrics that are preserved under projective transformations. The newly formulated weakly generalized Douglas-Weyl $(W-G D W)$ equation facilitates the generalization of generalized Douglas-Weyl $(G D W)$-metrics into the broader category of $W-G D W$-metrics, which encompasses all $G D W$-metrics. Within this class, there are also two additional subclasses: generalized weakly-Weyl metrics, characterized by a milder form of Weyl curvature, and generalized $\tilde{D}$-metrics, defined by a less strict version of Douglas curvature. The paper provides a comprehensive overview of these generalized class of Finsler metrics and elucidates their properties, supported by detailed examples.

New classes of Finsler metrics: The birth of new projective invatiant

TL;DR

The paper introduces a new projective invariant in Finsler geometry, the weakly generalized Douglas-Weyl (W-GDW) metrics, defined by the equation tildeD_j^i_{kl|0} + lambda F tildeD_j^i_{kl} = U_{jkl} y^i, and shows this class is closed under projective changes while containing all GDW metrics. It identifies two new subclasses, generalized tilde-D metrics and generalized weakly-Weyl metrics, and proves that every GDW-metric is a W-GDW-metric, while also deriving nontrivial conditions under which a regular (α,β)-metric becomes a W-GDW-metric without being GDW. The work connects these new invariants to existing projective invariants (Weyl, Douglas) and clarifies hierarchical relationships via lemmas, propositions, and examples. Overall, it broadens the landscape of projective invariants in Finsler geometry and provides a unified framework for analyzing curvature-based properties under projective transformations, with implications for αβ-metrics and their curvature theories.

Abstract

This paper presents a pioneering projective invariant in Finsler geometry, introducing a new class of Finsler metrics that are preserved under projective transformations. The newly formulated weakly generalized Douglas-Weyl equation facilitates the generalization of generalized Douglas-Weyl -metrics into the broader category of -metrics, which encompasses all -metrics. Within this class, there are also two additional subclasses: generalized weakly-Weyl metrics, characterized by a milder form of Weyl curvature, and generalized -metrics, defined by a less strict version of Douglas curvature. The paper provides a comprehensive overview of these generalized class of Finsler metrics and elucidates their properties, supported by detailed examples.

Paper Structure

This paper contains 7 sections, 15 theorems, 131 equations, 1 figure.

Table of Contents

  1. Abstract
  2. Introduction
  3. Preliminaries
  4. The birth of new projective tensor
  5. Weakly Generalized Douglas-Weyl Finsler metrics
  6. Generalized $\tilde{D}$-metrics
  7. Generalized Weakly-Weyl Finsler metrics

Key Result

Lemma 3.1

Sh2 Let $F$ and $\bar{F}$ be two projectively equivalent Finsler metrics on $M$. The Riemann curvatures are related by where Here $P_{|k}$ denotes the covariant derivative of projective factor $P$ with respect to $\bar{F}$.

Figures (1)

  • Figure 1: The diagram illustrates the relationships between different classes of Finsler metrics. The blue circle represents the class of Weyl metrics, denoted as Weyl. The red circle represents the class of Douglas metrics, denoted as D. The green ellipse encompasses the class of generalized weakly-Weyl metrics, denoted as G. W-Weyl. The purple ellipse represents the class of generalized $\tilde{D}$-metrics, denoted as G. $\tilde{D}$. The orange ellipse denotes the class of generalized Douglas-Weyl metrics, denoted as GDW. The gray ellipse represents the class of weakly generalized Douglas-Weyl metrics, denoted as W-GDW.

Theorems & Definitions (25)

  • Lemma 3.1
  • Lemma 3.2
  • Theorem 4.1
  • proof
  • Proposition 4.2
  • proof
  • Lemma 4.3
  • proof
  • Theorem 4.4
  • proof
  • ...and 15 more