Topic
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 article, the Ricci curvature in the direction of a certain vector field is greater than or equal to (n − 1)λ, forcing the vector field to be isometric to the n-sphere Sn(λ).
Abstract: In this paper, we consider an n-dimensional compact Riemannian manifold (M,g) of constant scalar curvature and show that the presence of a non-Killing conformal vector field ξ on M that is also an eigenvector of the Laplacian operator acting on smooth vector fields with eigenvalue λ together with a condition on Ricci curvature of M, that the Ricci curvature in the direction of a certain vector field is greater than or equal to (n − 1)λ, forces M to be isometric to the n-sphere Sn(λ).
32 citations
TL;DR: In this article, the authors consider a Riemannian metric in an open subset of the d-dimensional Euclidean space and assume that its RiemANN curvature tensor vanishes.
Abstract: We consider a Riemannian metric in an open subset of the d-dimensional Euclidean space and assume that its Riemann curvature tensor vanishes. If the metric is of class C2, a classical theorem in differential geometry asserts that the Riemannian space is locally isometrically immersed in the d-dimensional Euclidean space. We establish that if the metric belongs to the Sobolev space W1,∞ and its Riemann curvature tensor vanishes in the space of distributions, then the Riemannian space is still locally isometrically immersed in the d-dimensional Euclidean space.
32 citations
TL;DR: In this article, the authors generalized the conformal field theory developed for a Riemann sphere by Belavin, Polyakov, and Zamolodchikov to any RiemANN surface and studied the correlation functions of the energy-momentum tensor with local fields.
32 citations
TL;DR: In this paper, the authors established an optimal comparison result on the heat kernel of the conformal Laplacian on open manifolds with nonnegative Ricci curvature, and identified a class of functions with the following property.
Abstract: By establishing an optimal comparison result on the heat kernel of the conformal Laplacian on open manifolds with nonnegative Ricci curvature, (a) we show that many manifolds with positive scalar curvature do not possess conformal metrics with scalar curvature bounded below by a positive constant; (b) we identify a class of functions with the following property: If the manifold has a scalar curvature in this class, then there exists a complete conformal metric whose scalar curvature is any given function in this class. This class is optimal in some sense; (c) we have identified all manifolds with nonnegative Ricci curvature, which are “uniformly” conformal to manifolds with zero scalar curvature. Even in the Euclidean case, we obtain a necessary and sufficient condition under which the main existence results in [Ni1] and [KN] on prescribing nonnegative scalar curvature will hold. This condition had been sought in several papers in the last two decades.
32 citations
TL;DR: In this article, the authors derived a formula for the spectral action of Dirac operators on 4-dimensional manifolds with totally anti-symmetric torsion and deduced the Lagrangian for the Standard Model of particle physics in presence of torsions from the Chamseddine-Connes Dirac operator.
Abstract: We derive a formula for the gravitational part of the spectral action for Dirac operators on 4-dimensional manifolds with totally anti-symmetric torsion. We find that the torsion becomes dynamical and couples to the traceless part of the Riemann curvature tensor. Finally we deduce the Lagrangian for the Standard Model of particle physics in presence of torsion from the Chamseddine-Connes Dirac operator.
32 citations