Special anisotropic conformal changes of conic pseudo-Finsler surfaces
Nabil L. Youssef, Ebtsam H. Taha, A. A. Kotb, S. G. Elgendi
TL;DR
The paper addresses how direction-dependent (anisotropic) conformal changes affect curvature and invariant tensors on conic pseudo-Finsler surfaces. It defines $C$-anisotropic, horizontal $C$-anisotropic, vertical $C$-anisotropic, and $\phi T$-conditions, derives transformation rules for the modified Berwald frame, main scalar $\mathcal{I}$, and geodesic spray, and analyzes special cases including when the conformal factor equals the main scalar. Key contributions include criteria for invariance and projective relations under anisotropic conformal changes, the result that vertical $C$-anisotropic changes force a Riemannian metric, and connections between Landsberg and Berwald properties via $\phi T$-conditions, illustrated by a Finslerian Schwarzschild–de Sitter example. The work provides a concrete framework for constructing non-Riemannian metrics with prescribed symmetries and demonstrates the relevance of these transformations in Finsler gravity, supported by a comprehensive equivalence table for the various anisotropic changes.
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 $φT$-condition, the horizontal $φT$-condition, and the vertical $φT$-condition. The first two conditions reduce to the $\boldsymbolσ T$-condition when the conformal factor relies solely on a positional variable. We demonstrate that, under the vertical $φT$-condition change, every Landsberg surface is Berwaldian. Thus, the vertical $φT$-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.