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.
@article{MZM_2020_107_6_a0,
author = {A. V. Aminova and M. N. Sabitova},
title = {The {General} {Solution} of the {Eisenhart} {Equation}},
journal = {Matemati\v{c}eskie zametki},
pages = {803--816},
publisher = {mathdoc},
volume = {107},
number = {6},
year = {2020},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/MZM_2020_107_6_a0/}
}
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/