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: To study C(0)a priori estimates for solutions to certain complex Monge-Ampère equations, a coherent sheaf of ideals is introduced and it is shown that it satisfies various global algebrogeometric conditions, including a cohomology vanishing theorem.
Abstract: To study C0a priori estimates for solutions to certain complex Monge—Ampere equations, I introduce a coherent sheaf of ideals and show that it satisfies various global algebrogeometric conditions, including a cohomology vanishing theorem. This technique is used to establish the existence of Kahler-Einstein metrics of positive scalar curvature on a very large class of compact complex manifolds.
105 citations
Journal Article•
TL;DR: In this paper, two additional structures on X have been extensively studied: the Riemannian structure and the triangulation of X giving rise to combinatorial or polyhedral topology.
Abstract: Let X be a C ∞ closed manifold of dimension N. Two additional structures on X have been extensively studied. One is the Riemannian structure giving rise to Riemannian geometry and the other is the triangulation of X giving rise to combinatorial or polyhedral topology.
105 citations
01 Jan 1974
105 citations
TL;DR: Among all conformal classes of Riemannian metrics on the CP, the Fubini-Study metric has the largest Yamabe constant as discussed by the authors, which is proved by perturbations of the Seiberg-Witten equations, which yields new results on the total scalar curvature of almost Kahler 4-manifolds.
Abstract: Among all conformal classes of Riemannian metrics on ${\Bbb CP}_2$, that of the Fubini-Study metric is shown to have the largest Yamabe constant. The proof, which involves perturbations of the Seiberg-Witten equations, also yields new results on the total scalar curvature of almost-K\"ahler 4-manifolds.
105 citations
TL;DR: A geometrical framework for double field theory in which generalized Riemann and torsion tensors are defined without reference to a particular basis is introduced and it is shown that it contains the conventional Ricci tensor and scalar curvature but not the full Riem Mann tensor.
Abstract: We introduce a geometrical framework for double field theory in which generalized Riemann and torsion tensors are defined without reference to a particular basis. This invariant geometry provides a unifying framework for the frame-like and metric-like formulations developed before. We discuss the relation to generalized geometry and give an “index-free” proof of the algebraic Bianchi identity. Finally, we analyze to what extent the generalized Riemann tensor encodes the curvatures of Riemannian geometry. We show that it contains the conventional Ricci tensor and scalar curvature but not the full Riemann tensor, suggesting the possibility of a further extension of this framework.
105 citations