Anisotropic conformal change of conic pseudo-Finsler surfaces, I

TL;DR

This paper investigates anisotropic conformal transformations of conic pseudo-Finsler surfaces, deriving conditions for metric preservation, analyzing geometric object transformations, and exploring invariants and special cases.

Contribution

It provides necessary and sufficient conditions for anisotropic conformal changes to preserve pseudo-Finsler structures and studies their effects on geometric properties and invariants.

Findings

Derived tensorial change of inverse metric tensor using modified Berwald frame.

Identified conditions under which the transformed Finsler surface remains pseudo-Finsler.

Found invariants and conditions for flatness under anisotropic conformal transformations.

Abstract

The present work is devoted to investigate anisotropic conformal transformation of conic pseudo-Finsler surfaces $(M,F)$, that is, $ F(x,y)\longmapsto \overline{F}(x,y)=e^{\phi(x,y)}F(x,y)$, where the function $\phi(x,y)$ depends on both position $x$ and direction $y$, contrary to the ordinary (isotropic) conformal transformation which depends on position only. If $F$ is a pseudo-Finsler metric, the above transformation does not yield necessarily a pseudo-Finsler metric. Consequently, we find out necessary and sufficient condition for a (conic) pseudo-Finsler surface $(M,F)$ to be transformed to a (conic) pseudo-Finsler surface $(M,\overline{F})$ under the transformation $\overline{F}=e^{\phi(x,y)}F$. In general dimension, it is extremely difficult to find the anisotropic conformal change of the inverse metric tensor in a tensorial form. However, by using the modified Berwald frame on a Finsler surface, we obtain the change of the components of the inverse metric tensor in a tensorial form. This progress enables us to study the transformation of the Finslerian geometric objects and the geometric properties associated with the transformed Finsler function $\overline{F}$. In contrast to isotropic conformal transformation, we have a non-homothetic conformal factor $\phi(x,y)$ that preserves the geodesic spray. Also, we find out some invariant geometric objects under the anisotropic conformal change. Furthermore, we investigate a sufficient condition for $\overline{F}$ to be dually flat or/and projectively flat. Finally, we study some special cases of the conformal factor $\phi(x,y)$. Various examples are provided whenever the situation needs.