On canonical F-planar mappings of spaces with affine connection

Abstract

In this paper we study the theory of F-planar mappings of spaces with affine connection. We obtained condition, which preserved the curvature tensor. We also studied canonical F-planar mappings of space with affine connection onto symmetric spaces. In this case, the main equations have the partial differential Cauchy type form in covariant derivatives. We got the set of substantial real parameters on which depends the general solution of that PDE's system.

Invariants for F-Planar Mappings of Symmetric Affine Connection Spaces

“…where a and b are functions of t. [2,3,6,7,10] if any F -planar curve in A N is transformed to an F -planar curve in A N by the mapping f .…”

“…The F -planar mapping f transforms the affinor F i j to F i j . We will stay focused on F -planar mappings which preserve the F -structure [2,3,6,7,10]. In this case, the next equation holds (1.9) F i j = aF i j + bδ i j , for scalar functions a and b.…”

“…The F -planar mappings of symmetric affine connection spaces are involved by J. Mikeš and his research group [2][3][4][5][6][7]9,10]. This research is continued with F -planar mappings of non-symmetric affine connection spaces [11,16].…”

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

“…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.…”

“…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].…”