Anisotropic conformal change of conic pseudo-Finsler surfaces, II
Nabil L. Youssef, S. G. Elgendi, A. A. Kotb, Ebtsam H. Taha
TL;DR
This work extends anisotropic conformal changes on two-dimensional conic pseudo-Finsler surfaces by deriving transformation laws for Berwald, Landsberg, Douglas, and T-tensors under $\overline{F}=e^{\phi}F$. It provides necessary and sufficient conditions for anisotropic conformal changes to yield Berwald, Landsberg, or Douglas metrics, including when a Riemannian metric becomes Berwaldian and when the transformed metric is metrizable by a base Riemannian metric via a symmetric tensor $M^i_{jk}$. The results recover isotropic conformal theory as a special case and reveal that Szabó’s metrization theorem does not generalize to conic pseudo-Finsler surfaces, illustrated by a counterexample. The paper also offers criteria for preserving the $T$-condition and for anisotropic flatness, locally Minkowski structure, and Douglas properties under anisotropic conformal changes, together with explicit formulas in terms of the fundamental invariants $P$, $Q$, and the main scalar $\mathcal{I}$. Overall, the work clarifies how anisotropic conformal deformations generate non-Riemannian geometries from Riemannian ones and delineates the precise geometric conditions governing Berwald/Landsberg/Douglas behavior in this setting.
Abstract
This paper is a continuation of our investigation of the anisotropic conformal change of a conic pseudo-Finsler surface $(M,F)$, namely, the change $\overline{F}(x,y)=e^{φ(x,y)}F(x,y)$ \cite{first paper}. We obtain the relationship between some important geometric objects of $F $ and their corresponding objects of $\overline{F}$, such as Berwald, Landsberg and Douglas tensors, as well as the T-tensor. In contrast to isotropic conformal transformation, under an anisotropic conformal transformation, we find out the necessary and sufficient conditions for a Riemannian surface to be anisotropically conformal transformed to Berwald or Landsberg or Douglas surfaces. Consequently, we determine under what condition the geodesic spray of a two-dimensional pseudo-Berwald metric $\overline{F}$ is Riemann metrizable by a two-dimensional pseudo-Riemannian metric $F$. We show an example of a conformal transformation of a Riemannian metric $F$ that is not geodesically equivalent to a Riemannian metric but is instead Berwaldian. Also, we determine the necessary and sufficient conditions for $F $ to be anisotropically conformally flat (i.e., $\overline{F}$ is Minkowskian). Moreover, we identify the required conditions for preserving the $T$-condition under an anisotropic conformal change. Finally, we establish the necessary conditions for a Riemannian metric to be anisotropically conformal to a Douglas metric.