Anisotropic conformal change of conic pseudo-Finsler surfaces, I

Abstract The present work is devoted to investigate anisotropic conformal transformation of conic pseudo-Finsler surfaces ( M , F ), that is, F ( x , y ) ⟼ F ― ( x , y ) = e ϕ ( x , y ) F ( x , y ) , where the function ϕ ( 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 , F ― ) under the transformation F ― = e ϕ ( 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 F ― . In contrast to isotropic conformal transformation, we have a non-homothetic conformal factor ϕ ( 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 F ― to be dually flat or/and projectively flat. Finally, we study some special cases of the conformal factor ϕ ( x , y ) . Various examples are provided whenever the situation needs.