On conformal transformations of general \((\alpha,\beta)\)-metrics
A general \((\alpha,\beta)\)-metric is a Finsler metric expressed in the form: \(F=\alpha\phi(b^2,s), s=\frac{\beta}{\alpha}\), where \(\alpha:=\sqrt{a_{ij}(x)y^i y^j}\) is a Riemannian metric, \(\beta:=b_i(x)y^i\) is a 1-form, \(b:=||\beta||_\alpha\) and \(\phi(b^2,s)\) is a positive smooth function. An \((\alpha,\beta)\)-metric is a general \((\alpha,\beta)\)-metric, but the converse is not true in general. Two Finsler metrics \(F\) and \(\widetilde{F}\) on a manifold \(M\) are conformally related if \(\widetilde{F}=e^{\sigma(x)}F\), where \(\sigma=\sigma(x)\) is a scalar function on \(M\) called conformal factor. A Finsler metric \(F\) said to have isotropic \(S\)-curvature if its \(S\)-curvature takes the form \(\textbf{S}=(n+1)cF\), where \(c=c(x)\) is a scalar function on \(M\). In this paper the authors characterize the conformal transformations between two general \((\alpha,\beta)\)-metrics. They also discuss conformally related general \((\alpha,\beta)\)-metrics with isotropic S-curvature. The main results in this respect are the following: Let two non-Riemannian general \((\alpha,\beta)\)-metrics \(F=\alpha\phi(b^2,s)\) and \(\widetilde{F}=\alpha\widetilde{\phi}(b^2,s)\) be conformally related on a manifold \(M\) of dimension \(n\ge 3\) with the conformal factor \(\sigma(b^2)\). (1) Assume that \(F\) is a Douglas metric and \(\phi_1\phi_2-\phi\phi_{12}=0\). Then, \(\widetilde{F}\) is a Douglas metric if and only if the conformal transformation is a homothety. (2) Assume that \(F\) is of isotropic \(S\)-curvature. Then \(\widetilde{F}\) is of isotropic \(S\)-curvature if and only if the conformal transformation is a homothety.
scientific article; zbMATH DE number 7377357
scientific article