Ricci decomposition

About: Ricci decomposition is a research topic. Over the lifetime, 1972 publications have been published within this topic receiving 45295 citations.

Positivity of General Superenergy Tensors

Abstract: Several of the most important results in general relativity require or assume positivity properties of certain tensors. The positive energy theorem and the singularity theorems make assumptions about the energy-momentum tensor and Ricci tensor respectively. Positivity of the Bel–Robinson tensor is needed in the proof of the global stability of Minkowski spacetime. Senovilla has recently presented a procedure of how to construct a superenergy tensor from any tensor. For a Maxwell field or a scalar field the procedure yields the usual energy-momentum tensor, for the Weyl tensor and the Riemann tensor one obtains the Bel–Robinson tensor and Bel tensor respectively. In general, by considering any tensor as an r-fold n 1,…,n r )-form, one constructs a rank 2r superenergy tensor from it. By using spinor methods, we prove that the contraction of any such superenergy tensor with 2r future-pointing vectors is non-negative. We refer to this as the dominant superenergy property and it generalizes several previous positivity results obtained for certain tensors as well as it provides a unified way of treating them. Some more examples are given and applications discussed.

Indefinite Almost Paracontact Metric Manifolds

Abstract: We introduce the concept of ()-almost paracontact manifolds, and in particular, of ()-para-Sasakian manifolds. Several examples are presented. Some typical identities for curvature tensor and Ricci tensor of ()-para Sasakian manifolds are obtained. We prove that if a semi-Riemannian manifold is one of flat, proper recurrent or proper Ricci-recurrent, then it cannot admit an ()-para Sasakian structure. We show that, for an ()-para Sasakian manifold, the conditions of being symmetric, semi-symmetric, or of constant sectional curvature are all identical. It is shown that a symmetric spacelike (resp., timelike) ()-para Sasakian manifold is locally isometric to a pseudohyperbolic space (resp., pseudosphere ). At last, it is proved that for an ()-para Sasakian manifold the conditions of being Ricci-semi-symmetric, Ricci-symmetric, and Einstein are all identical.