DEVIATION EQUATIONS OF SYNGE AND SCHILD OVER $({\bar L}_n,g)$-SPACES

Abstract

The deviation equation of Synge and Schild has been investigated over differentiable manifolds with contravariant and covariant affine connections (whose components differ not only by sign) and metric [[Formula: see text]-spaces]. It is shown that the condition £ξu=0 for obtaining this equation is only a sufficient (but not necessary) condition. By means of a nonisotropic (nonnull) vector field u [g(u,u)=e≠0] and the metric hu, orthogonal to it, a projected deviation equation of Synge and Schild has been obtained for the orthogonal to u vector field ξ⊥ and its square L2=g(ξ⊥, ξ⊥). For a given nonisotropic, auto-parallel and normalized vector field u this equation could have some simple solutions.