The General Solution of the Eisenhart Equation and Projective Motions of Pseudo-Riemannian Manifolds

The solution of the Eisenhart equation for pseudo-Riemannian manifolds (M-n,g) of arbitrary signature and any dimension is obtained. Thereby, pseudo-Riemannianh-spaces (i.e., spaces admitting nontrivial solutionsh not equal cgof the Eisenhart equation) of all possible types determined by the Segre characteristic chi of the bilinear formhare found. Necessary and sufficient conditions for the existence of an infinitesimal projective transformation in (M-n,g) are given. The curvature 2-form of a (rigid)h-space of type chi= {r(1), horizontal ellipsis ,r(k)} is calculated and necessary and sufficient conditions for this space to have constant curvature are obtained.