Geodesic
The General Solution of the Eisenhart Equation
Matematičeskie zametki, Tome 107 (2020) no. 6, pp. 803-816

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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.
Keywords: Eisenhart equation, $h$-space, projective motion, curvature. 
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     author = {A. V. Aminova and M. N. Sabitova},
     title = {The {General} {Solution} of the {Eisenhart} {Equation}},
     journal = {Matemati\v{c}eskie zametki},
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     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2020_107_6_a0/}
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A. V. Aminova; M. N. Sabitova. The General Solution of the Eisenhart Equation. Matematičeskie zametki, Tome 107 (2020) no. 6, pp. 803-816. http://geodesic.mathdoc.fr/item/MZM_2020_107_6_a0/