Conformally geodesic mappings satisfying a certain initial condition

Hana Chudá and Josef Mikeš

Dept. of Mathematics, Faculty of Applied Informatics, Tomas Bata University T.G. Masaryka 5555, 760 01 Zlín, Czech Republic
Department of Algebra and Geometry, Faculty of Science, Palacky University, 17. listopadu 12, 779 00 Olomouc, Czech Republic


Abstract: In this paper we study conformally geodesic mappings between pseudo-Riemannian manifolds $(M, g)$ and $(\bar{M}, \bar{g})$, i.e. mappings $f\colon M \rightarrow \bar{M}$ satisfying $f = f_1 \circ f_2 \circ f_3$, where $f_1, f_3$ are conformal mappings and $f_2$ is a geodesic mapping. Suppose that the initial condition $f^* \bar{g} = k g$ is satisfied at a point $x_0 \in M$ and that at this point the conformal Weyl tensor does not vanish. We prove that then $f$ is necessarily conformal.

AMSclassification: primary 53B20; secondary 53B30, 53C21.

Keywords: conformal mappings, geodesic mappings, conformally geodesic mappings.