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### 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.

** Keywords:** Kaluza Ñ Klein, gravity, electromagnetic field, differential geometry, five-dimensional pseudo-Riemannian manifold,\(h\)-space \(H_{221}\), \(H_{32}\), \(H_{41}\), \(H_{5}\), systems of partial differential equations, non-homothetical projective motion, the Killing equations, projective Lie algebra.

** UDC:** 514.763

** PACS:** 11.10.Kk, 04.50.+h, 04.50.-h, 02.40-k, 02.20.Sv

** DOI:** 10.17238/issn2226-8812.2023.2.4-27

*Please cite this article in English as:*

Aminova A.V., Khakimov D.R. Projective symmetries of five-dimensional spaces. Space, Time and Fundamental Interactions, 2023, no. 2, pp. 4-27.