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Beyond the Generalized Douglas Weyl spaces

Nasrin Sadeghzadeh, Najmeh Sajjadi Moghadam

TL;DR

The paper analyzes the hierarchy of projective-invariant Finsler metrics centered on Generalized Douglas–Weyl (GDW) metrics, clarifying how Douglas, Weyl, $R$-quadratic, and $PR$-quadratic sub-classes fit within GDW. It formalizes GDW via $D_j{^i}_{kl|m}y^m=T_{jkl}y^i$ and introduces the bar-D metric class, showing that Douglas ⊂ bar-D ⊂ GDW with many GDW metrics not belonging to Douglas or bar-D. The work proves inclusion relations among sub-classes (R-quadratic and PR-quadratic inside GDW) and provides concrete examples (including Randers-type and constant-flag-curvature spaces) to illustrate non-Douglas GDW metrics and the circumstances under which GDW reduces to Douglas. It also relates PR-quadratic properties to $S$-curvature and $R$-quadratic conditions, offering insights into the structure and separations among these projective invariants.

Abstract

This paper gives new insights into the class of Generalized Douglas Weyl ($GDW$)-metrics. This projective invariant class of Finsler metrics, contains some well-known Finsler metrics such as Douglas, Weyl and $R$-quadratic metrics. Here, some new sub-classes of $GDW$-metrics are constructed and considered as the explicit Finsler metrics. Many illustrative and interesting examples are presented.

Beyond the Generalized Douglas Weyl spaces

TL;DR

The paper analyzes the hierarchy of projective-invariant Finsler metrics centered on Generalized Douglas–Weyl (GDW) metrics, clarifying how Douglas, Weyl, -quadratic, and -quadratic sub-classes fit within GDW. It formalizes GDW via and introduces the bar-D metric class, showing that Douglas ⊂ bar-D ⊂ GDW with many GDW metrics not belonging to Douglas or bar-D. The work proves inclusion relations among sub-classes (R-quadratic and PR-quadratic inside GDW) and provides concrete examples (including Randers-type and constant-flag-curvature spaces) to illustrate non-Douglas GDW metrics and the circumstances under which GDW reduces to Douglas. It also relates PR-quadratic properties to -curvature and -quadratic conditions, offering insights into the structure and separations among these projective invariants.

Abstract

This paper gives new insights into the class of Generalized Douglas Weyl ()-metrics. This projective invariant class of Finsler metrics, contains some well-known Finsler metrics such as Douglas, Weyl and -quadratic metrics. Here, some new sub-classes of -metrics are constructed and considered as the explicit Finsler metrics. Many illustrative and interesting examples are presented.

Paper Structure

This paper contains 5 sections, 9 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalized Douglas Weyl Finsler metrics
  4. Douglas curvature of Finsler metrics
  5. $PR$-quadratic Finsler metrics

Key Result

Lemma 2.1

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 $\bar{F}$.

Theorems & Definitions (17)

  • Lemma 2.1
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Example 3.4
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Corollary 3.3
  • Proposition 3.4
  • ...and 7 more