Intrinsic Study of a Special Anisotropic Conformal Transformation

Abstract

In this paper, we present a coordinate-free investigation of a special anisotropic conformal transformation of a Finsler function \(L\). Specifically, for a Finsler metric \((M, L)\) and a one-form \(\mathfrak{B}\), we consider the transformation
\[\widetilde{L}(x, \dot x) = e^{\sigma(x,  \dot x)} L(x,  \dot x) = e^{\frac{\mathfrak{B}(x, \dot x)}{L(x,  \dot x)}} L(x,  \dot x).\]
We examine various geometric objects associated with the transformed metric \(\widetilde{L}(x,  \dot x)\) in terms of the corresponding objects of the original metric \(L\). In particular, we derive the expressions for the metric tensor, Cartan tensor, and other related geometric quantities for \(\widetilde{L}\), and we characterize the conditions under which the metric tensor of \(\widetilde{L}\) is non-degenerate. To further explore geometric structures such as the geodesic spray, Barthel connection, and Berwald connection of \(\widetilde{L}(x,  \dot x)\), we restrict the one-form \(\mathfrak{B}\) to one induced by a concurrent \(\pi\)-vector field. Under this assumption, we compute the curvature of the Barthel connection associated with \(\widetilde{L}\). An explicit example is also provided.