First quadratic integral of geodesics with certain initial conditions

Abstract

In this paper we investigate first quadratic integrals of geodesics a of n-dimensionaly (pseudo-) Riemannian spaces Vn with metric g satisfying at a finite number of points x following conditions ax = f(x) gx. It comes out that even under this conditions it holds that first quadratic integrals of geodesics is trivial, i.e. a = const g.