2025/03/10 by Youssef, Nabil L., Elgendi, S. G., Kotb, A. A. +1 · 1 citation
#Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2503.07842
This paper is a continuation of our investigation of the anisotropic conformal change of a conic pseudo-Finsler surface (M,F), namely, the change F(x,y)=eϕ(x,y)F(x,y) \citefirst paper. We obtain the relationship between some important geometric objects of F and their corresponding objects of 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 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., 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.