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 paper, a quintom model of dark energy with a single scalar field T given by a Lagrangian which inspired by tachyonic Lagrangians in string theory is considered.
97 citations
96 citations
TL;DR: In this paper, it was shown that the necessary and sufficient conditions for rotationally symmetric solutions are: R > 0 and R′(r) changes signs in the region where R is positive.
Abstract: on S for n ≥ 3. In the case R is rotationally symmetric, the well-known Kazdan-Warner condition implies that a necessary condition for (1) to have a solution is: R > 0 somewhere and R′(r) changes signs. Then, (a) is this a sufficient condition? (b) If not, what are the necessary and sufficient conditions? These have been open problems for decades. In Chen & Li, 1995, we gave question (a) a negative answer. We showed that a necessary condition for (1) to have a solution is: R′(r) changes signs in the region where R is positive. (2)
96 citations
TL;DR: For n = 2, the Yamabe conjecture was proved as discussed by the authors, which states that there exist metrics that are pointwise conformal to g and have constant Gauss curvature.
Abstract: Let (M, g) be an n-dimensional compact smooth Riemannian manifold (without boundary). For n=2, we know from the uniformization theorem of Poinca% that there exist metrics that are pointwise conformal to g and have constant Gauss curvature. For n~>3, the well-known Yamabe conjecture states that there exist metrics which are pointwise conformal to g and have constant scalar curvature. The Yamabe conjecture is proved through the work of Yamabe [65], Trudinger [58], Aubin [2] and Schoen [53]. The Yamabe and related problems have attracted much attention in the last 30 years or so, see, e.g., [57], [3] and the references therein. Important methods and techniques in overcoming loss of compactness have been developed in such studies, which also play important roles in the research of other areas of mathematics. For n~>3, let ~=u4/(n-2)g, where u is some positive function on M. The scalar curvature RO of ~ can be calculated as
96 citations
TL;DR: In this article, it was shown that the universality of the Einstein equations and Komar's energy-momentum complex also extends to this case (modulo a conformal transformation of the metric).
Abstract: It has been shown recently that, in the first-order (Palatini) formalism, there is universality of the Einstein equations and the Komar energy-momentum complex, in the sense that for a generic nonlinear Lagrangian depending only on the scalar curvature of a metric and a torsionless connection one always gets the Einstein equations and Komar's expression for the energy-momentum complex. In this paper a similar analysis (also within the framework of the first-order formalism) is performed for all nonlinear Lagrangians depending on the (symmetrized) Ricci square invariant. The main result is that the universality of the Einstein equations and Komar's energy-momentum complex also extends to this case (modulo a conformal transformation of the metric).
96 citations