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

Abstract

In the paper we consider almost geodesic mappings of the first type of spaces with affine connections onto generalized 2-Ricci-symmetric spaces, generalized 3-Ricci-symmetric spaces, and generalized m-Ricci-symmetric spaces. In either case the main equations for the mappings are obtained as a closed system of linear differential equations of Cauchy type in the covariant derivatives. The obtained results extend an amount of research produced by N.S. Sinyukov, V.E. Berezovski, J. Mikeš.

Almost geodesic mappings of type π1* of spaces with affine connection

“…This result was generalized for Ricci-Codazzi spaces with affine connection and for Riemannian spaces in [15]. For mappings of general symmetric spaces with affine connection the system of differential equations in the Cauchy form were found in works [16].…”

“…Contracting (5) for and , it is easy to see that it holds (7) where (8) Taking account of (7), the formulas (5) are expressible in the form (9) where and are the Weyl tensors of projective curvature of the spaces and respectively.…”

“…Let and be affine coordinate systems in the spaces and respectively. A point mapping (16) where , , are some constant, , defines the almost geodesic mapping of the space onto the space . By direct calculation it is readily shown that the components of the deformation tensor in the coordinate system are given all the other components being zero.…”

“…In the theory of geodesic mappings and their generalizations many basic results were formulated as a system of differential equations in Cauchy form, see [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. For almost geodesic mappings a similar result for special Ricci-Codazzi Riemannian spaces is formulated in Sinyukov monograph [1].…”

“…The concept of almost geodesic mappings of type of spaces with affine torsion-free connections was first introduced in [16]. These mappings are a special case of type almost geodesic mappings which were introduced by N. S. Sinyukov in [1].…”