42 research outputs found
Linear Connections and Curvature Tensors in the Geometry of Parallelizable Manifolds
In this paper we discuss curvature tensors in the context of Absolute
Parallelism geometry. Different curvature tensors are expressed in a compact
form in terms of the torsion tensor of the canonical connection. Using the
Bianchi identities some other identities are derived from the expressions
obtained. These identities, in turn, are used to reveal some of the properties
satisfied by an intriguing fourth order tensor which we refer to as Wanas
tensor. A further condition on the canonical connection is imposed, assuming it
is semi-symmetric. The formulae thus obtained, together with other formulae
(Ricci tensors and scalar curvatures of the different connections admitted by
the space) are calculated under this additional assumption. Considering a
specific form of the semi-symmetric connection causes all nonvanishing
curvature tensors to coincide, up to a constant, with the Wanas tensor.
Physical aspects of some of the geometric objects considered are mentioned.Comment: 16 pages LaTeX file, Changed title, Changed content, Added
references, Physical features stresse
Weakly Z symmetric manifolds
We introduce a new kind of Riemannian manifold that includes weakly-, pseudo-
and pseudo projective- Ricci symmetric manifolds. The manifold is defined
through a generalization of the so called Z tensor; it is named "weakly Z
symmetric" and denoted by (WZS)_n. If the Z tensor is singular we give
conditions for the existence of a proper concircular vector. For non singular Z
tensor, we study the closedness property of the associated covectors and give
sufficient conditions for the existence of a proper concircular vector in the
conformally harmonic case, and the general form of the Ricci tensor. For
conformally flat (WZS)_n manifolds, we derive the local form of the metric
tensor.Comment: 13 page