On equitorsion concircular tensors of generalized Riemannian spaces

Abstract

In this paper we consider concircular vector fields of manifolds with non-symmetric metric tensor. The subject of our paper is an equitorsion concircular mapping. A mapping f : GR N → GR N is an equitorsion if the torsion tensors of the spaces GR N and GR N are equal. For an equitorsion concircular mapping of two generalized Riemannian spaces GR N and GR N , we obtain some invariant curvature tensors of this mapping Z θ , θ = 1, 2,. .. , 5, given by equations (3.14, 3.21, 3.28, 3.31, 3.38). These quantities are generalizations of the concircular tensor Z given by equation (2.5). i jk = ip Γ p. jk respectively, where (i j) = (i j) −1 and i j denotes symmetrization with division of the indices i and j. Generally the generalized Christoffel symbols 2010 Mathematics Subject Classification. Primary 53B05;

Conformal Equitorsion and Concircular Transformations in a Generalized Riemannian Space

“…For five independent tensors K θ in [19], the invariants Z θ were found, which are different from the invariantsZ θ in the present paper (see Remark 3.1, at the end). Compare e.g.,Z 1 from the present paper and…”

Generalized Weyl conformal curvature tensor of generalized Riemannian space

“…ln |g| ,j and T α jα = 0, (1.5) for g = det[g ij ] = 0. The Riemannian space R N endowed with the affine connection coefficients γ i jk is called the associated space of GR N [15,16,18,21,24]. Since the associated space R N is usual Riemannian space, there exists only one kind of covariant differentiation with regard to the affine connection of this space:…”

Some invariants of conformal mappings of a generalized Riemannian space

“…for the tensor ψ i as above and the tensor ξ i jk anti-symmetric by indices j and k. The Weyl conformal curvature tensor C i jmn is generalized for conformal mappings which preserve the torsion tensor T i jk (also called the equitorsion conformal mappings) in [22,23]. Invariants of random conformal mappings of equidistant Riemannian spaces are obtained in [19].…”

“…Definition 1.1. [19,22,23] An N-dimensional manifold M N endowed with a metric tensor G ij , G ij G ji , is the generalized Riemannian space GR N .…”