Scalar curvature

About: Scalar curvature is a research topic. Over the lifetime, 12701 publications have been published within this topic receiving 296040 citations. The topic is also known as: Ricci scalar.

A Generalization of Hawking's Black Hole Topology Theorem to Higher Dimensions

Abstract: Hawking’s theorem on the topology of black holes asserts that cross sections of the event horizon in 4-dimensional asymptotically flat stationary black hole spacetimes obeying the dominant energy condition are topologically 2-spheres. This conclusion extends to outer apparent horizons in spacetimes that are not necessarily stationary. In this paper we obtain a natural generalization of Hawking’s results to higher dimensions by showing that cross sections of the event horizon (in the stationary case) and outer apparent horizons (in the general case) are of positive Yamabe type, i.e., admit metrics of positive scalar curvature. This implies many well-known restrictions on the topology, and is consistent with recent examples of five dimensional stationary black hole spacetimes with horizon topology S2 × S1. The proof is inspired by previous work of Schoen and Yau on the existence of solutions to the Jang equation (but does not make direct use of that equation).

Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes

Abstract: There has been much interest among differential geometers in finding relationships between curvature and topology of Riemannian manifolds. For the most part, efforts have been directed towards understanding the implications of positive or negative sectional curvature, Ricci curvature, or scalar curvature, but there are other hypotheses on the curvature which also deserve investigation. In this article, we will consider the topological implications of a new curvature assumption, positive curvature on totally isotropic two-planes, and we will prove via the Sacks-Uhlenbeck theory of minimal two-spheres that a compact simply connected Riemannian manifold of dimension at least four, with positive curvature on totally isotropic two-planes is homeomorphic to a sphere. As corollaries, we will obtain a proof via minimal two-spheres (for Riemannian manifolds of dimension at least four) of the Berger-Klingenberg-Rauch-Toponogov sphere theorem, strengthened to pointwise quarter-pinching, as well as a proof that every simply connected compact Riemannian manifold with positive curvature operators is homeomorphic to a sphere. Let M be an n-dimensional Riemannian manifold with tangent space TpM at the point p E M. Recall that the curvature operator at p is the self-adjoint

Some remarks on G2-structures

Abstract: This article consists of loosely related remarks about the geometry of G2structures on 7-manifolds, some of which are based on unpublished joint work with two other people: F. Reese Harvey and Steven Altschuler. After some preliminary background information about the group G2 and its representation theory, a set of techniques is introduced for calculating the differential invariants of G2-structures and the rest of the article is applications of these results. Some of the results that may be of interest are as follows: First, a formula is derived for the scalar curvature and Ricci curvature of a G2structure in terms of its torsion and covariant derivatives with respect to the ‘natural connection’ (as opposed to the Levi-Civita connection) associated to a G2-structure. When the fundamental 3-form of the G2-structure is closed, this formula implies, in particular, that the scalar curvature of the underlying metric is nonpositive and vanishes if and only if the structure is torsion-free. These formulae are also used to generalize a recent result of Cleyton and Ivanov [3] about the nonexistence of closed Einstein G2-structures (other than the Ricci-flat ones) on compact 7-manifolds to a nonexistence result for closed G2-structures whose Ricci tensor is too tightly pinched. Second, some discussion is given of the geometry of the first and second order invariants of G2-structures in terms of the representation theory of G2. Third, some formulae are derived for closed solutions of the Laplacian flow that specify how various related quantities, such as the torsion and the metric, evolve with the flow. These may be useful in studying convergence or long-time existence for given initial data. Some of this work was subsumed in the work of Hitchin [12] and Joyce [14]. I am making it available now mainly because of interest expressed by others in seeing these results written up since they do not seem to have all made it into the literature. Received by the editors Februay 01, 2005. 1991 Mathematics Subject Classification. 53C10, 53C29.