Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, Vol. 26, No. 2, pp. 337-341 (2010)

On geodesic mappings with certain initial conditions

Josef Mikes and Hana Chudá

Palacky University of Olomouc and Thomas Bata University

Abstract: In this paper we studied geodesic mappings $f$ between (pseudo-)Riemannian spaces $V_n$ and $\bar{V_n}$ with the following initial condition $\bar g(f(x_o))=k\cdot g(x_o)$, where $g$ and $\bar g$ are the metrics of $V_n$ and $\bar{V_n}$.

We proved that if the Weyl tensor of the projective curvature is not vanishing at the point $x_o\in\vn$ then $f$ is homothetic.

Keywords: geodesic mapping, initial conditions, (pseudo-) Riemannian space

Classification (MSC2000): 53B20; 53B30, 53C21

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