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, Bahri and Brezis [BB], Bahri [B], and Schoen [S2-3] showed that a large class of compact Riemannian manifolds with boundary are conformally equivalent to one with constant scalar curvature and zero mean curvature on the boundary.
Abstract: Let (M,g) be an n dimensional compact, smooth, Riemannian manifold without boundary. For n = 2, the Uniformization Theorem of Poincare says that there exist metrics on M which are pointwise conformal to g and have constant Gauss curvature. For n > 3, the well known Yamabe conjecture states that there exist metrics on M which are pointwise conformal to g and have constant scalar curvature. The Yamabe conjecture has been proved through the work of Yamabe [Y], Trudinger [T], Aubin [A], and Schoen [SI]. See Lee and Parker [LP] for a survey. See also Bahri and Brezis [BB], Bahri [B], and Schoen [S2-3] for works on the problem and related ones. Analogues of the Yamabe problem for compact Riemannian manifolds with boundary have been studied by Cherrier, Escobar, and others. In particular, Escobar proved in [E2] that a large class of compact Riemannian manifolds with boundary are conformally equivalent to one with constant scalar curvature and zero mean curvature on the boundary. See also [E3][E5] for related results. From now on in the paper, (M, g) denotes some smooth compact n dimensional Riemannian manifold with boundary, unless we specify otherwise. We use M to denote the interior of M, and dM the boundary of M. We use n — 2 d n — 2 La to denote An—c(n)Ra, where c(n) is — —, BQ to denote ——I—-—hQ, y y * 4(n -1) y du 2 y where u is the outward unit normal on dM with respect to 5, and hg to denote the mean curvature of dM with respect to the inner normal (balls in R have positive mean curvatures).
101 citations
101 citations
TL;DR: In this article, the authors correct the derivation of this equation of motion and comment on some of its consequences, since they noticed that the authors missed an essential term which has consequences in the equations of motion of test particles.
Abstract: In Phys. Rev. D 84, 024020 (2011) Harko, Lobo, Nojiri and Odintsov presented a modified theory of gravitation, $f(R,T)$ gravity, where the gravitational Lagrangian is given by an arbitrary function of the Ricci scalar and of the trace of the stress-energy tensor. In this Comment we correct the conservation equation of the stress-energy tensor, since we noticed that the authors missed an essential term which has consequences in the equation of motion of test particles. We thus correct the derivation of this equation of motion and comment on some of its consequences.
101 citations
TL;DR: In this article, a new method is presented for assigning distributional curvature, in an invariant manner, to a spacetime of low differentiability, using the techniques of Colombeau's ''new generalized functions''.
Abstract: A new method is presented for assigning distributional curvature, in an invariant manner, to a spacetime of low differentiability, using the techniques of Colombeau's `new generalized functions'. The method is applied to show that the scalar curvature density of a cone is equivalent to a delta function. The same is true under small enough perturbations.
100 citations
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TL;DR: In this paper, a search strategy for asymptotic safety for a simplified version of gravity in four dimensions using the renormalization group is proposed. But the search strategy is based on the Ricci scalar and curvature invariants become increasingly irrelevant with increasing mass dimension.
Abstract: A search strategy for asymptotic safety is put forward and tested for a simplified version of gravity in four dimensions using the renormalization group. Taking the action to be a high-order polynomial of the Ricci scalar, a self-consistent ultraviolet fixed point is found where curvature invariants become increasingly irrelevant with increasing mass dimension. Intriguingly, universal scaling exponents take near-Gaussian values despite the presence of residual interactions. Asymptotic safety of metric gravity would seem in reach if this pattern carries over to the full theory.
100 citations