# Article

Ufa Mathematical Journal
Volume 5, Number 3, pp. 40-52

# On some special solutions of Eisenhart equation

Zakirova Z.Kh.

DOI:10.13108/2013-5-3-40

In this note we study a $6$-dimensional pseudo-Riemannian space $V^6(g_{ij})$ with the signature $[++----]$, which admits projective motions, i.e., continuous transformation groups preserving geodesics. A general method of determining pseudo-Riemannian spaces admitting some nonhomothetic projective group $G_r$ was developed by A.V. Aminova. A.V. Aminova classified all Lorentzian manifolds of dimension greater than three admitting nonhomothetic projective or affine infinitesimal transformations. The problem of classification is not solved for pseudo-Riemannian spaces with arbitrary signature. In order to find a pseudo-Riemannian space admitting a nonhomothetic infinitesimal projective transformation, one has to integrate Eisenhart equation \begin{equation*} h_{ij,k}=2g_{ij} \varphi_{,k}+g_{ik} \varphi_{,j}+ g_{jk} \varphi_{,i}. \end{equation*} Pseudo-Riemannian manifolds for which there exist nontrivial solutions $h_{ij}\ne cg_{ij}$ to the Eisenhart equation are called {\it $h$-spaces}. It is known that the problem of describing such spaces depends on the type of the $h$-space, i.e., on the type of the bilinear form $L_{X}g_{ij}$ determined by the characteristic of the $\lambda$-matrix $(h_{ij}-\lambda g_{ij})$. The number of possible types depends on the dimension and the signature of an $h$-space In this work we find the metric and determine quadratic first integrals of the corresponding geodesic lines equations for 6-dimensional $h$-spaces of the type $[(21\ldots1)(21\ldots1)\ldots(1\ldots1)]$.