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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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Journal ArticleDOI
TL;DR: In this article, the geometry of B-manifolds is studied, and some properties of Riemannian curvature tensors of paraholomorphic B-mansifolds are given.

66 citations

Journal ArticleDOI
TL;DR: In this paper, a Weyl conformal factor is used to perturb the standard volume form and obtain the Laplacian that encodes the local geometric information of noncommutative 4-tori T^4-θ.
Abstract: In this paper we study the curved geometry of noncommutative 4-tori T^4_θ. We use a Weyl conformal factor to perturb the standard volume form and obtain the Laplacian that encodes the local geometric information. We use Connes' pseudodifferential calculus to explicitly compute the terms in the small time heat kernel expansion of the perturbed Laplacian which correspond to the volume and scalar curvature of T^4_θ. We establish the analogue of Weyl's law, define a noncommutative residue, prove the analogue of Connes' trace theorem, and find explicit formulas for the local functions that describe the scalar curvature of T^4_θ. We also study the analogue of the Einstein-Hilbert action for these spaces and show that metrics with constant scalar curvature are critical for this action.

66 citations

Journal ArticleDOI
TL;DR: In the simplest case of ordinary dilaton gravity the well kown problem of removing the Schwarzschild singularity by a field redefinition is clarified, and the equivalent dilaton one exhibits solutions which are asymptotically flat in special ranges of the parameters.
Abstract: Two dimensional gravity with torsion is proved to be equivalent to special types of generalized 2d dilaton gravity. E.g. in one version, the dilaton field is shown to be expressible by the extra scalar curvature, constructed for an independent Lorentz connection corresponding to a nontrivial torsion. Elimination of that dilaton field yields an equivalent torsionless theory, nonpolynomial in curvature. These theories, although locally equivalent exhibit quite different global properties of the general solution. We discuss the example of a (torsionless) dilaton theory equivalent to the R 2 + T 2 –model. Each global solution of this model is shown to split into a set of global solutions of generalized dilaton gravity. In contrast to the theory with torsion the equivalent dilaton one exhibits solutions which are asymptotically flat in special ranges of the parameters. In the simplest case of ordinary dilaton gravity we clarify the well kown problem of removing the Schwarzschild singularity by a field redefinition.

66 citations

Journal ArticleDOI
TL;DR: In this article, the authors studied the butterfly effect in D-dimensional gravitational theories containing terms quadratic in Ricci scalar and Ricci tensor and observed that due to higher order derivatives in the corresponding equations of motion there are two butterfly velocities.
Abstract: We study butterfly effect in D-dimensional gravitational theories containing terms quadratic in Ricci scalar and Ricci tensor. One observes that due to higher order derivatives in the corresponding equations of motion there are two butterfly velocities. The velocities are determined by the dimension of operators whose sources are provided by the metric. The three dimensional TMG model is also studied where we get two butterfly velocities at generic point of the moduli space of parameters. At critical point two velocities coincide.

65 citations

Journal ArticleDOI
TL;DR: In this paper, the first-order curvature equation coincides with the Monge-Ampere equation, and the second-order equation with the first order curvature equations of order.
Abstract: Solvability conditions for curvature equations of order which are sufficient, and almost necessary, are obtained, and theorems concerning the existence of solutions in , , , are proved. The first-order curvature equation coincides with the curvature equation of order , and the curvature equation of order with the Monge-Ampere equation.Bibliography: 18 titles.

65 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023268
2022536
2021505
2020448
2019424
2018433