Special anisotropic conformal changes of conic pseudo-Finsler surfaces

TL;DR

This paper investigates special anisotropic conformal transformations of conic pseudo-Finsler surfaces, revealing their properties, characterizations, and implications for Riemannian and Finslerian geometries, including applications to a Schwarzschild-de Sitter solution.

Contribution

It introduces new types of anisotropic conformal changes in conic pseudo-Finsler surfaces and explores their geometric properties and conditions, including applications to specific solutions.

Findings

Vertical $C$-anisotropic conformal changes characterize Riemannian surfaces.

Vertical $oldsymbol{ ext{φ} T}$-condition implies Landsberg surfaces are Berwaldian.

Application to a Finslerian Schwarzschild-de Sitter solution with spherical symmetry.

Abstract

This study presents many special anisotropic conformal changes of a conic pseudo-Finsler surface $(M,F)$, such as $C$-anisotropic and horizontal $C$-anisotropic conformal transformations, which reduce to $C$-conformal when the conformal factor is solely position-dependent. Furthermore, we present vertical $C$-anisotropic conformal changes and demonstrate that they are characterized by the property of $(M,F)$ being Riemannian. Additionally, we examine the anisotropic conformal transformation that fulfils the $\phi T$-condition, the horizontal $\phi T$-condition, and the vertical $\phi T$-condition. The first two conditions reduce to the $\boldsymbol{\sigma} T$-condition when the conformal factor relies solely on a positional variable. We demonstrate that, under the vertical $\phi T$-condition change, every Landsberg surface is Berwaldian. Thus, the vertical $\phi T$-condition is equivalent to the $T$-condition. Furthermore, we examine the scenario when the anisotropic conformal factor becomes the main scalar of the non-Riemannian surface $(M,F)$. We present an example of a Finslerian Schwarzschild-de Sitter solution having Finslerian spherical symmetry and apply our results to it.