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Riemann curvature tensor
About: Riemann curvature tensor is a research topic. Over the lifetime, 6248 publications have been published within this topic receiving 138871 citations. The topic is also known as: Riemann–Christoffel tensor.
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TL;DR: In this paper, a local test function construction was used to seattle the most cases left by Escobar's and Marques's works and reduce the remaining case to the positive mass theorem.
Abstract: In 1992, motivated by Riemann mapping theorem, Escobar considered a version of Yamabe problem on manifolds of dimension n greater than 2 with boundary. The problem consists in finding a conformal metric such that the scalar curvature is zero and the mean curvature is constant on the boundary. By using a local test function construction, we are able to seattle the most cases left by Escobar's and Marques's works. Moreover, we reduce the remaining case to the positive mass theorem. In this proof, we use the method developed in previous works by Brendle and by Brendle and the author.
34 citations
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TL;DR: The existence of closed hypersurfaces of prescribed scalar curvature in globally hyperbolic Lorentzian manifolds is proved provided there are barriers as discussed by the authors, which is not the case here.
Abstract: The existence of closed hypersurfaces of prescribed scalar curvature in globally hyperbolic Lorentzian manifolds is proved provided there are barriers.
33 citations
TL;DR: In this article, the authors defined the total scalar curve plus total mean curve functional on the space of Riemannian metrics of a smooth compact manifold with boundary and characterized its critical points.
Abstract: The Total Scalar Curvature plus Total Mean Curvature functional is defined on the space of Riemannian metrics of a smooth compact manifold with boundary. We characterize its critical points restricted to spaces of Riemannian metrics satisfying various volume and area constraints, when the dimension of the manifold is n ≥ 3. In addition, we compute the second variation of said functional at critical points and exhibit directions in which it is positive, negative or zero. These results generalize to manifolds with boundary, well known results that hold in the case of manifolds without boundary.
33 citations
TL;DR: In this article, it was shown that if the Ricci curvature of a closed Riemannian manifold with a RiemANNIAN foliation of codimension q is isometric to the standard q-sphere of constant sectional curvature a, then the smallest positive eigenvalue A of the Laplacian satisfies A > an.
Abstract: The standard Lichnerowicz comparison theorem states that if the Ricci curvature of a closed, Riemannian n-manifold M satisfies Ric (X, X) > a(n - 1) |X| 2 for every X E TM for some fixed a > 0, then the smallest positive eigenvalue A of the Laplacian satisfies A > an. The Obata theorem states that equality occurs if and only if M is isometric to the standard n-sphere of constant sectional curvature a. In this paper, we prove that if M is a closed Riemannian manifold with a Riemannian foliation of codimension q, and if the normal Ricci curvature satisfies Ric⊥ (X,X) ≥ a (q - 1) |X| 2 for every X in the normal bundle for some fixed a > 0, then the smallest eigenvalue λ B of the basic Laplacian satisfies λ B > aq. In addition, if equality occurs, then the leaf space is isometric to the space of orbits of a discrete subgroup of O (q) acting on the standard q-sphere of constant sectional curvature a. We also prove a result about bundle-like metrics on foliations: On any Riemannian foliation with bundle-like metric, there exists a bundle-like metric for which the mean curvature is basic and the basic Laplacian for the new metric is the same as that of the original metric.
33 citations
TL;DR: In this article, it was shown that the number of ends of a complete manifold with Ricci curvature nonnegative outside a compact set is finite and an explicit upper bound for the number is given.
Abstract: We consider complete manifolds with Ricci curvature nonnegative outside a compact set and prove that the number of ends of such a manifold is finite and in particular, we give an explicit upper bound for the number.
33 citations