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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, a solution of the well-known question: which Fano manifolds admit Kahler-Einstein metrics was announced. But the condition they end up using is essentially as proposed by Tian [20] and which we now recall (see also [14] for the equivalence with a priori stronger definitions).
Abstract: This is a note to announce a solution of the well-known question: which Fano manifolds admit Kahler–Einstein metrics? The idea that the appropriate condition should be in terms of “algebro-geometric stability” was proposed about 20 years ago by Yau [21, 22] (partly by analogy with the “Kobayashi–Hitchin correspondence” in the case of holomorphic bundles). For more on the history of this problem please see our forthcoming full paper. Over the years various different notions of stability have been discussed in the literature, both in the Fano/Kahler–Einstein case and in the more general situation of constant scalar curvature Kahler metrics on polarized manifolds. But the condition we end up using is essentially as proposed by Tian [20] and which we now recall (see also [14] for the equivalence with a priori stronger definitions).

66 citations

Journal ArticleDOI
TL;DR: In this article, the authors discuss linear connections and curvature tensors in the context of geometry of parallelizable manifolds (or absolute parallelism geometry) using Bianchi identities.

66 citations

Journal ArticleDOI
TL;DR: In this paper, counterexamples to Min-Oo's conjecture in dimension n \geq 3 were constructed for a Riemannian manifold M of dimension n whose boundary is totally geodesic and is isometric to the standard sphere S n-1.
Abstract: Consider a compact Riemannian manifold M of dimension n whose boundary \partial M is totally geodesic and is isometric to the standard sphere S^{n-1}. A natural conjecture of Min-Oo asserts that if the scalar curvature of M is at least n(n-1), then M is isometric to the hemisphere S_+^n equipped with its standard metric. This conjecture is inspired by the positive mass theorem in general relativity, and has been verified in many special cases. In this paper, we construct counterexamples to Min-Oo's conjecture in dimension n \geq 3.

66 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that physically reasonable solutions of the field equations based on an R2 lagrangian are possible, however, experimental predictions of such a theory are at variance with observations.
Abstract: It is shown that physically reasonable solutions of the field equations based on an R2 lagrangian are possible. (R is the scalar curvature.) However, it is shown that experimental predictions of such a theory are at variance with observations. The most general quadratic lagrangian is also considered and it is shown that the R2 term must dominate thus invalidating gravitational equations based on a general quadratic lagrangian.

66 citations

Journal ArticleDOI
TL;DR: In this paper, a geometrical approach to statistical thermodynamics is proposed, and it is shown that any r-parameter generalised Gibbs distribution leads to a Riemannian metric of parameter space.
Abstract: A geometrical approach to statistical thermodynamics is proposed. It is shown that any r-parameter generalised Gibbs distribution leads to a Riemannian metric of parameter space. The components of the metric tensor are represented by second moments of stochastic variables. The scalar curvature R, as a geometrical invariant, is a function of the second and third moments, so is strictly connected with fluctuations of the system. In the case of a real gas, R is positive and tends to infinity as the system approaches the critical point. In the case of an ideal gas, R=0. The obtained results, and the results of the authors previous work, suggests that for a wide class of models R tends to + infinity near the critical point. They treat R as a measure of the stability of the system. They propose some sort of statistical principle: only such models may be accepted for which R tends to infinity if the system is approaching the critical point. It is shown that, if this criterion is adopted for a class of models for which the scaling hypothesis holds, then they obtain the new inequalities for the critical indices. These inequalities are in good agreement with model calculations and experiment.

66 citations


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