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Finsler surfaces with vanishing $T$-tensor

Salah G. Elgendi

Abstract

In this paper, for Finsler surfaces, we prove that the T-condition and $σT$-condition coincide. For higher dimensions $n\geq 3$, we illustrate by an example that the T-condition and $σT$-condition are not equivalent. We show that the non-homothetic conformal change of a Berwald (resp. a Landsberg) surface is Berwaldian (resp. Landsbergian) if and only if the $σT$-condition is satisfied. By solving the Landsberg's PDE, we classify all Finsler surfaces satisfying the T-condition, or equivalently the $σT$-condition. Some examples are provided and studied.

Finsler surfaces with vanishing $T$-tensor

Abstract

In this paper, for Finsler surfaces, we prove that the T-condition and -condition coincide. For higher dimensions , we illustrate by an example that the T-condition and -condition are not equivalent. We show that the non-homothetic conformal change of a Berwald (resp. a Landsberg) surface is Berwaldian (resp. Landsbergian) if and only if the -condition is satisfied. By solving the Landsberg's PDE, we classify all Finsler surfaces satisfying the T-condition, or equivalently the -condition. Some examples are provided and studied.
Paper Structure (6 sections, 14 theorems, 83 equations)

This paper contains 6 sections, 14 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. T-condition and $\sigma T$-condition
  4. Finsler surfaces
  5. Conformal Change
  6. Finsler surfaces satisfying the T-condition

Key Result

Lemma 3.2

Matsumoto For Finsler surface $(M,F)$, we have the following associated geometric objects: where $I$ is a $0$-homogeneous function in $y$ and called the main scalar of the manifold $(M,F)$.

Theorems & Definitions (29)

  • Definition 2.1
  • Definition 2.2
  • Example 1
  • Example 2
  • Lemma 3.2
  • Definition 3.3
  • Theorem 3.5
  • proof
  • Lemma 3.6
  • Lemma 3.7
  • ...and 19 more