Topic
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.
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TL;DR: In this article, the supremum of the sectional curvature of a Riemannian manifold is estimated in terms of the curvatures of the manifold and the radius of the ball.
Abstract: 1. Statement of the result. All manifolds considered in this paper shall be connected, of class C"' (smooth), and dimension at least 2. Im- mersions will also be smooth, and of codimension at least 1. If M and M are Riemannian manifolds and (p: M - M is an isometric immersion with the property that sp(M) lies in a ball, we intend to estimate the supremum of the sectional curvature K of M in terms of the sectional curvature K of M and the radius of the ball. This is our result: THEOREM A. Let Ml' be a complete Riemannian manifold whose scalar curvature is bounded below; let M" +q be a Riemannian manifold with q c n -1, and Bx a closed normal ball in Mt4+q, of radius X. Sup- pose Sp:M" - j M"+q is an isometric immersion with the property that
130 citations
TL;DR: In this paper, Michor et al. studied the geometry of (M, G) by using the ideas developed in [Michor, 1980] and used variational principles to compute geodesics as the curves in M minimizing the energy functional.
Abstract: In this paper we study the geometry of (M, G) by using the ideas developed in [Michor, 1980]. With that differentiable structure on M it is possible to use variational principles and so we start in section 2 by computing geodesics as the curves in M minimizing the energy functional. From the geodesic equation, the covariant derivative of the Levi-Civita connection can be obtained, and that provides a direct method for computing the curvature of the manifold. Christoffel symbol and curvature turn out to be pointwise in M and so, although the mappings involved in the definition of the Ricci tensor and the scalar curvature have no trace, in our case we can define the concepts of ”Ricci like curvature” and ”scalar like curvature”. The pointwise character mentioned above allows us in section 3, to solve explicitly the geodesic equation and to obtain the domain of definition of the
129 citations
TL;DR: In this article, the authors consider the evolution of the graph of a Riemannian manifold by the mean curvature flow and show that the flow exists smoothly for all time.
Abstract: Let f:Σ1↦Σ2 be a map between compact Riemannian manifolds of constant curvature. This article considers the evolution of the graph of f in Σ1×Σ2 by the mean curvature flow. Under suitable conditions on the curvature of Σ1 and Σ2 and the differential of the initial map, we show that the flow exists smoothly for all time. At each instant t, the flow remains the graph of a map ft and ft converges to a constant map as t approaches infinity. This also provides a regularity estimate for Lipschitz initial data.
129 citations
TL;DR: In this article, the existence of a Noether vector seems to rule the presence of higher-order corrections of gravity in a class of theories of gravity where the Ricci scalar R and its d'Alembertian □ R are present.
Abstract: Higher-order corrections of the Einstein–Hilbert action of general relativity can be recovered by imposing the existence of a Noether symmetry to a class of theories of gravity where the Ricci scalar R and its d'Alembertian □ R are present. In several cases, it is possible to get exact cosmological solutions or, at least, to simplify the dynamics by recovering constants of motion. The main result is that a Noether vector seems to rule the presence of higher-order corrections of gravity.
129 citations
TL;DR: The global structure of McVittie's solution representing a point mass embedded in a spatially flat Robertson-Walker universe is investigated in this article, where the scalar curvature singularity at proper radius R = 2m, where m (constant) is the Schwarzschild mass, and the apparent horizon which surrounds it are studied.
Abstract: The global structure of McVittie's solution representing a point mass embedded in a spatially flat Robertson-Walker universe is investigated. The scalar curvature singularity at proper radius R=2m, where m (constant) is the Schwarzschild mass, and the apparent horizon which surrounds it are studied. The conformal diagram for the spacetime is obtained via a qualitative analysis of the radial null geodesics. Particular attention is paid to the physical interpretation of this spacetime; previous work on this issue is reviewed, and to how recent quasi-local definitions of black and white holes relate to this spacetime.
129 citations