SPACE, TIME AND FUNDAMENTAL INTERACTIONS

PROJECTIVE SYMMETRIES OF FIVE-DIMENSIONAL SPACES

Aminova A.V., Khakimov D.R.

A review of invariant-group methods in 5-dimensional theories of electromagnetic, gravitational and other physical fields is presented. The symmetries of the five-dimensional curved spaces in the form of Lie groups of infinitesimal transformations, in particular, in the form of projective motions which preserve geodesics are discussed. The 5-dimensional rigid $h$-spaces $H_{221}$, $H_{32}$, $H_{41}$ and $H_{5}$, i.e. pseudo-Riemannian manifolds $(M^5, g)$ of arbitrary signature with (non-degenerate) Segre characteristic $\chi = \{r_1, ..., r_k\}$, $r_1, ..., r_k \in N$, $r_1 + ... + r_k = 5$, and real eigenvalues of the Lie derivative $L_{Xg}$ of the metric $g$ in the direction of the infinitesimal transformation $X$ are investigated, which admit (non-homothetic) infinitesimal projective and affine transformations, and for each of them the structure of the corresponding maximal projective and affine Lie algebras are determined; the classification of $h$-spaces $H_{221}$ of type {221} on maximal Lie algebras of projective and affine transformations, wider than the Lie algebras of homotheties, is obtained.