Geodesic Mappings of Spaces with Affine Connections onto Generalized Symmetric and Ricci-Symmetric Spaces

Abstract

In the paper, we consider geodesic mappings of spaces with an affine connections onto generalized symmetric and Ricci-symmetric spaces. In particular, we studied in detail geodesic mappings of spaces with an affine connections onto 2-, 3-, and m- (Ricci-) symmetric spaces. These spaces play an important role in the General Theory of Relativity. The main results we obtained were generalized to a case of geodesic mappings of spaces with an affine connection onto (Ricci-) symmetric spaces. The main equations of the mappings were obtained as closed mixed systems of PDEs of the Cauchy type in covariant form. For the systems, we have found the maximum number of essential parameters which the solutions depend on. Any m- (Ricci-) symmetric spaces (m≥1) are geodesically mapped onto many spaces with an affine connection. We can call these spaces projectivelly m- (Ricci-) symmetric spaces and for them there exist above-mentioned nontrivial solutions.

Canonical Almost Geodesic Mappings of the First Type of Spaces with Affine Connections onto Generalized m-Ricci-Symmetric Spaces

“…In this paper, we study canonical almost geodesic mappings of the first type of spaces with an affine connection onto generalized m-Ricci-symmetric spaces. These spaces are the essential generalization of the spaces considered in [24]. Ricci tensor is well known from the Einstein equation that describes gravitational fields.…”

“…Obviously, in the space A n the Equations ( 3), ( 20)-( 22), ( 32) and ( 33) form a closed system of PDEs of Cauchy type with respect to the functions P h ij (x), a ij (x), R h ijk (x), R h ijkm (x), a ijk (x), a ijkm (x). Also the functions must satisfy the algebraic conditions (24).…”

“…Let us differentiate covariantly (m − 3) times the Equation ( 27) with respect to the connection of the space A n . Then express in the left-hand side the covariant derivatives with respect to the connection of the space A n in terms of the covariant derivatives with respect to the connection of the space A n , using the formula [24]…”

“…Sobchuk [19,20], N.Y. Yablonskaya [21,22], V.E. Berezovski, J. Mikeš [14,15,[23][24][25][26][27][28][29][30][31][32][33][34][35], O. Belova, J. Mikeš, K. Strambach [36,37], M.S. Stankovič, Lj.S.…”

“…Let us note that in works [7,8,24,[46][47][48][49][50] were studied geodesic and holomorphically projective mappings of above mentioned spaces. For the other mappings the main equations are obtained as closed systems of PDEs of Cauchy type, see [5,6,[51][52][53][54].…”