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-Riemannian h -spaces (i.e., spaces admitting nontrivial solutions h ≠ cg of the Eisenhart equation) of all possible types determined by the Segrè characteristic χ of the bilinear form h are 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 χ = { r 1 , …, r k } is calculated and necessary and sufficient conditions for this space to have constant curvature are obtained.