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 intrig...

Let us define a curvature invariant of the order k as a scalar polynomial constructed from gαβ, the Riemann tensor Rαβγδ, and covariant derivatives of the Riemann tensor up to the order k. According to this definition, the Ricci curvature scalar R or the Kretschmann curvature scalar RαβγδR αβγδ are curvature invariants of the order zero and Rαβγδ;εR αβγδ;ε is a curvature invariant of the order ...

We derive matter collineations for some static spherically symmetric spacetimes and compare the results with Killing, Ricci and Curvature symmetries. We conclude that matter and Ricci collineations are not, in general, the same.

Abstract We introduce the notion of rigidity for harmonic-Ricci solitons and provide some characterizations rigidity, generalizing known results Ricci solitons. In complete case we restrict to steady shrinking gradient solitons, while in compact treat general without further assumptions. show that can be traced back vanishing certain modified curvature tensors take into account geometry a Riema...

In [H4], Hamilton determined a sharp tensor Harnack inequality for complete solutions of the Ricci flow with non-negative curvature operator. This Harnack inequality is of critical importance to the understanding of singularities of the Ricci flow, as is evident from its numerous applications in [H3], [H5], [H6], and [H7]. Moreover, according to Hamilton, the discovery of a Harnack inequality i...

We consider Bel–Robinson-like higher derivative conserved two-index tensors H µν in simple matter models, following a recently suggested Maxwell field version. In flat space, we show that they are essentially equivalent to the true stress-tensors. In curved Ricci-flat backgrounds it is possible to redefine H µν so as to overcome non-commutativity of covariant derivatives, and maintain conservat...