Lie algebras of projective motions of five-dimensional pseudo-Riemannian spaces. IV. Structure of projective and affine Lie algebras of five-dimensional rigid $h$-spaces

This work is devoted to the problem of studying multidimensional pseudo-Riemannian manifolds that admit Lie algebras of infinitesimal projective (in particular, affine) transformations, wider than Lie algebras of infinitesimal homotheties. Such manifolds have numerous geometric and physical applications. This paper is the fourth part of the work. The first part: Itogi Nauki i Tekhniki. Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. — 2022. — 212. — P. 10–29. The second part: Itogi Nauki i Tekhniki. Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. — 2022. — 213. — P. 10–37. The third part: Itogi Nauki i Tekhniki. Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. — 2022. — 214. — P. 3–20. The last part will be published in the next issue.