Ricci and Matter Collineations of Locally Rotationally Symmetric Space-Times

A new method is presented for the determination of Ricci Collineations (RC) and Matter Collineations (MC) of a given spacetime, in the cases where the Ricci tensor and the energy momentum tensor are non-degenerate and have a similar form with the metric. This method reduces the problem of finding the RCs and the MCs to that of determining the KVs whereas at the same time uses already known results on the motions of the metric. We employ this method to determine all hypersurface homogeneous locally rotationally symmetric spacetimes, which admit proper RCs and MCs. We also give the corresponding collineation vectors. These results conclude a long due open problem, which has been considered many times partially in the literature.     

The Huber theorem for non-compact conformally flat manifolds

It was proved in 1957 by Huber that any complete surface with integrable Gauss curvature is conformally equivalent to a compact surface with a finite number of points removed. Counterexamples show that the curvature assumption must necessarily be strengthened in order to get an analogous conclusion in higher dimensions. We show in this paper that any non compact Riemannian manifold with finite -norm of the Ricci curvature satisfies Huber-type conclusions if either it is a conformal domain with volume growth controlled from above in a compact Riemannian manifold or if it is conformally flat of dimension 4 and a natural Sobolev inequality together with a mild scalar curvature decay assumption hold. We also get partial results in other dimensions. Received: April 14, 2000; revised version: March 20, 2001     

A compactness result for Kähler Ricci solitons

In this paper we prove a compactness result for compact Kähler Ricci gradient shrinking solitons. If (Mi,gi) is a sequence of Kähler Ricci solitons of real dimension n?4, whose curvatures have uniformly bounded Ln/2 norms, whose Ricci curvatures are uniformly bounded from below and μ(gi,1/2)?A (where μ is Perelman's functional), there is a subsequence (Mi,gi) converging to a compact orbifold (M_∞,g_∞) with finitely many isolated singularities, where g_∞ is a Kähler Ricci soliton metric in an orbifold sense (satisfies a soliton equation away from singular points and smoothly extends in some gauge to a metric satisfying Kähler Ricci soliton equation in a lifting around singular points).     

On quasi-conformally flat weakly Ricci symmetric manifolds

The object of the present paper is to study quasi-conformally flat weakly Ricci symmetric manifolds.      

Comment: Towards a Complete Classification of Spherically Symmetric Lorentzian Manifolds According to Their Ricci Collineations

General expressions for the components of the Ricci collineation vector are derived and the related constraints are obtained. These constraints are then solved to obtain Ricci collineations and the related constraints on the Ricci tensor components for all spacetime manifolds (degenerate or non-degenerate, diagonal or non-diagonal) admitting symmetries larger than so(3) and already known results are recovered. A complete solution is achieved for the spacetime manifolds admitting so(3) as the maximal symmetry group with non-degenerate and non diagonal Ricci tensor components. It is interesting to point out that there appear cases with finite number of Ricci collineations although the Ricci tensor is degenerate and also the cases with infinitely many Ricci collineations even in the case of non-degenerate Ricci tensor. Interestingly, it is found that the spacetime manifolds with so(3) as maximal symmetry group may admit two extra proper Ricci collineations, although they do not admit a G ₅ as the maximal symmetry group. Examples are provided which show and clarify some comments made by Camci et al. [Camci, U., and Branes, A. (2002). Class. Quantum Grav. 19, 393–404]. Theorems are proved which correct the earlier claims made in [Carot, J., Nunez, L. A., and Percoco, U. (1997). Gen. Relativ. Gravit. 29, 1223–1237; Contreras, G., Núñez, L. A., and Percolo, U. (2000). Gen. Relativ. Gravit. 32, 285–294].     

Canonical Connections and Algebraic Ricci Solitons of Three-dimensional Lorentzian Lie Groups

In this paper, the author computes canonical connections and KobayashiNomizu connections and their curvature on three-dimensional Lorentzian Lie groups with some product structure. He defines algebraic Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections. He classifies algebraic Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections on three-dimensional Lorentzian Lie groups with some product structure.     

Unification of electromagnetism and gravitation: The equations of electrodynamics derived from the Riemann tensor in general relativity

The equations of free-space electrodynamics are derived directly from the Riemann curvature tensor and the Bianchi identity of general relativity by contracting on two indices to give a novel antisymmetric Ricci tensor. Within a factore/h, this is the field-strength tensor G of free-space electrodynamics. The Bianchi identity for G describes free-space electrodynamics in a manner analogous to, but more general than, Maxwell's equations for electrodynamics, the critical difference being the existence in general and special relativity of the Evans-Vigier fieldB ⁽³⁾.     

The prescribed Ricci curvature problem on three‐dimensional unimodular Lie groups

Let G be a three‐dimensional unimodular Lie group, and let T be a left‐invariant symmetric (0,2)‐tensor field on G. We provide the necessary and sufficient conditions on T for the existence of a pair consisting of a left‐invariant Riemannian metric g and a positive constant c such that , where is the Ricci curvature of g. We also discuss the uniqueness of such pairs and show that, in most cases, there exists at most one positive constant c such that is solvable for some left‐invariant Riemannian metric g.     

On Gradient Solitons of the Ricci–Harmonic Flow

In this paper, we study gradient solitons to the Ricci flow coupled with harmonic map heat flow. We derive new identities on solitons similar to those on gradient solitons of the Ricci flow. When the soliton is compact, we get a classification result. We also discuss the relation with quasi-Einstein manifolds.