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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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TL;DR: In this article, it was shown that the Ricci curvature of a Riemannian metric is either everywhere positive or everywhere negative, and the divergence and curl both vanish.
Abstract: We shall prove theorems on nonexistence of certain types of vector fields on a compact manifold with a positive definite Riemannian metric whose Ricci curvature is either everywhere positive or everywhere negative. Actually we shall have some relaxations of the requirements both as to curvature and as to compactness. We shall deal with real spaces with a customary metric and with complex analytic spaces with an Hermitian metric. In the latter case we shall impose on the metric a certain restriction, first explicitly stated by E. Kaehler, which will be quite indispensable to our argument. In order to elucidate the rôle of this restriction we shall include a systematic introduction to the theory of Hermitian metric. For positive curvature we shall have the theorem that on a compact space there exists no vector field for which the divergence and curl both vanish. In the complex case there exists no vector field whatsoever whose covariant components are analytic functions in the complex parameters. If we only assume that the curvature is nonnegative, then there are some \"exceptional\" vector fields in directions of spatial flatness. A principal result will be the following theorem on meromorphic functions. If a complex space with positive curvature is covered by a finite number of neighborhoods, if a meromorphic functional element is defined in each neighborhood, and if the difference of any two meromorphic elements is holomorphic wherever the elements overlap, then there exists one meromorphic function on the space which differs by a holomorphic function from each meromorphic element given. In a previous paper this conclusion was drawn in the

296 citations

Journal ArticleDOI
TL;DR: In this paper, some characterizations of totally real submanifolds are given, and some classifications of complex space forms are obtained for complex analytic and totally real subsets of an almost Hermitian manifold.
Abstract: Complex analytic submanifolds and totally real submanifolds are two typical classes among all submanifolds of an almost Hermitian mani- fold. In this paper, some characterizations of totally real submanifolds are given. Moreover some classifications of totally real submanifolds in complex space forms are obtained.

294 citations

Journal ArticleDOI
TL;DR: In this article, the authors consider the minimization problem with constraints and show that the proximal point method in Euclidean space is naturally extended to solve that class of problems.
Abstract: In this paper we consider the minimization problem with constraints. We will show that if the set of constraints is a Riemannian manifold of nonpositive sectional curvature, and the objective function is convex in this manifold, then the proximal point method in Euclidean space is naturally extended to solve that class of problems. We will prove that the sequence generated by our method is well defined and converge to a minimizer point. In particular we show how tools of Riemannian geometry, more specifically the convex analysis in Riemannian manifolds, can be used to solve nonconvex constrained problem in Euclidean, space.

293 citations

Journal ArticleDOI
TL;DR: In this paper, the authors constructed a supergravity model whose scalar degrees of freedom arise from a chiral superfield and are solely a scalaron and an axion that is very heavy during the inflationary phase.

292 citations

Journal ArticleDOI
TL;DR: In this paper, the generalized Tanaka connection for contact Riemannian manifolds generalizing one for nondegenerate, integrable CR manifolds was defined, and the torsion and generalized Tanaka-Webster scalar curvature were defined properly.
Abstract: We define the generalized Tanaka connection for contact Riemannian manifolds generalizing one for nondegenerate, integrable CR manifolds. Then the torsion and the generalized Tanaka-Webster scalar curvature are defined properly. Furthermore, we define gauge transformations of contact Riemannian structure, and obtain an invariant under such transformations. Concerning the integral related to the invariant, we define a functional and study its first and second variational formulas. As an example, we study this functional on the unit sphere as a standard contact manifold.

291 citations


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