On geodesic mappings with certain initial conditions. (English) Zbl 1240.53029

Mikeš, Josef; Chudá, Hana

Acta Math. Acad. Paedagog. Nyházi. (N.S.) 26, No. 2, 337-341 (2010).

Summary: In this paper we studied geodesic mappings \(f\) between (pseudo-) Riemannian spaces \(V_n\) and \(\bar V_n\) with the initial condition \(\bar g(f(x_0)) =k \cdot g(x_0)\), 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 \(x_0\in V_n\) then \(f\) is homothetic.

MSC:

53B20	Local Riemannian geometry
53B30	Local differential geometry of Lorentz metrics, indefinite metrics
53C21	Methods of global Riemannian geometry, including PDE methods; curvature restrictions

Keywords:

geodesic mapping; initial conditions; (pseudo-) Riemannian space