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, it was shown that generalized gravity theories involving the curvature invariants of the Ricci tensor and the Riemann tensor are equivalent to multi-scalar-tensor gravities with four-derivative terms.
Abstract: We show that generalized gravity theories involving the curvature invariants of the Ricci tensor and the Riemann tensor as well as the Ricci scalar are equivalent to multi-scalar–tensor gravities with four-derivative terms. By expanding the action around a vacuum spacetime, the action is reduced to that of the Einstein gravity with four-derivative terms, and consequently there appears a massive spin-2 ghost in such generalized gravity theories in addition to a massive spin-0 field.
168 citations
13 Jun 2005
TL;DR: In this paper, a fast and robust method for detecting crest lines on surfaces approximated by dense triangle meshes is proposed, which is based on estimating the curvature tensor and curvature derivatives via local polynomial fitting.
Abstract: We propose a fast and robust method for detecting crest lines on surfaces approximated by dense triangle meshes. The crest lines, salient surface features defined via first- and second-order curvature derivatives, are widely used for shape matching and interrogation purposes. Their practical extraction is difficult because it requires good estimation of high-order surface derivatives. Our approach to the crest line detection is based on estimating the curvature tensor and curvature derivatives via local polynomial fitting.Since the crest lines are not defined in the surface regions where the surface focal set (caustic) degenerates, we introduce a new thresholding scheme which exploits interesting relationships between curvature extrema, the so-called MVS functional of Moreton and Sequin, and Dupin cyclides,An application of the crest lines to adaptive mesh simplification is also considered.
168 citations
TL;DR: In this paper, the notion of k-Ricci curvature of a Riemannian n-manifold was defined and sharp relations between the k-ricci curvatures and the shape operator were established.
Abstract: First we define the notion of k-Ricci curvature of a Riemannian n- manifold. Then we establish sharp relations between the k-Ricci curvature and the shape operator and also between the k-Ricci curvature and the squared mean cur- vature for a submanifold in a Riemannian space form with arbitrary codimension. Several applications of such relationships are also presented.
168 citations
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16 Apr 2009TL;DR: In this article, a rigidity theorem for Riemannian fibrations of flat spaces over compact bases is proved and a metric classification of compact four-dimensional manifolds of nonnegative curvature that admit totally geodesic RiemANNian foliations is given.
Abstract: We prove a rigidity theorem for Riemannian fibrations of flat spaces over compact bases and give a metric classification of compact four-dimensional manifolds of nonnegative curvature that admit totally geodesic Riemannian foliations.
167 citations
TL;DR: This paper shall define a conformal Killing tensor in another way and generalize some results about a conformAL Killing vector to the conformalkilling tensor.
Abstract: where pc is a certain vector field. Because we can easily show that a conformal Killing tensor in this sense is a Killing tensor, i.e., we have pc = 0. Thus this definition of conformal Killing tensor is meaningless. In this paper we shall define a conformal Killing tensor in another way and generalize some results about a conformal Killing vector to the conformal Killing tensor. The definition which we shall adopt is suggested by the following fact. A parallel vector field in the Euclidean space E induces a
167 citations