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 $h\ne cg$ of the Eisenhart equation) of all possible types determined by the Segrè characteristic $\chi$ 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 $\chi=\{r_1,\dots,r_k\}$ is calculated and necessary and sufficient conditions for this space to have constant curvature are obtained.