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Ricci decomposition

About: Ricci decomposition is a research topic. Over the lifetime, 1972 publications have been published within this topic receiving 45295 citations.


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TL;DR: In this paper, it was shown that a Ricci soliton is rigid if and only if the Weyl conformal tensor of the Ricci tensor is harmonic, assuming that the curvature tensor has at most exponential growth.
Abstract: We show that a compact Ricci soliton is rigid if and only if the Weyl conformal tensor is harmonic. In the complete noncompact case we prove the same result assuming that the curvature tensor has at most exponential growth and the Ricci tensor is bounded from below.

102 citations

Journal ArticleDOI
30 Aug 2007
TL;DR: In this article, it was shown that a complete Riemannian metric of nonnegative sectional curvature can be deformed to a metric of positive Ricci curvature.
Abstract: In this paper we address the question whether a complete Riemannian metric of nonnegative sectional curvature can be deformed to a metric of positive Ricci curvature. This problem came up implicitly in various recent new constructions for metrics with positive Ricci curvature. Grove and Ziller [GZ] showed that any compact cohomogeneity one manifold with finite fundamental group admits invariant metrics with positive Ricci curvature. The case that both non-regular orbits have codimension two is especially resilient. By earlier work of Grove and Ziller it has been known that these manifolds admit invariant nonnegatively curved metrics. However, in certain cases the Ricci curvature of these metrics is not positive at any point and hence they cannot apply the deformation theorem of Aubin [A] and Ehrlich [E]: a metric of nonnegative Ricci curvature is conformally equivalent to a metric with positive Ricci curvature if and only if the Ricci curvature is positive at some point. Similar problems arise in the work of Schwachhöfer and Tuschmann on quotient spaces [ST]. Our main result is: Theorem A. Let (Mn, g) be a compact Riemannian manifold with finite fundamental group and nonnegative sectional curvature. Then Mn admits a metric with positive Ricci curvature.

100 citations

Journal ArticleDOI
01 Jan 1990-Topology
TL;DR: In this article, Weinberger et al. showed that the Ricci curvature of a complete manifold is a sufficient condition for the existence of an almost nilpotent subgroup.

98 citations

Journal ArticleDOI
TL;DR: In this article, the Ricci tensor of a real hypersurface M in complex two-plane Grassmannians G 2 (C m +2 ) was derived from the equation of Gauss.
Abstract: . In this paper, flrst we introduce the full expression of thecurvature tensor of a real hypersurface M in complex two-plane Grass-mannians G 2 (C m +2 ) from the equation of Gauss and derive a new formulafor the Ricci tensor of M in G 2 (C m +2 ). Next we prove that there do notexist any Hopf real hypersurfaces in complex two-plane Grassmannians G 2 (C m +2 ) with parallel and commuting Ricci tensor. Finally we showthat there do not exist any Einstein Hopf hypersurfaces in G 2 (C m +2 ). IntroductionIn the geometry of real hypersurfaces in complex space forms or in quater-nionic space forms it can be easily checked that there do not exist any realhypersurfaces with parallel shape operator A by virtue of the equation of Co-dazzi.But if we consider a real hypersurface with parallel Ricci tensor S in suchspace forms, the proof of its non-existence is not so easy. In the class of Hopfhypersurfaces Kimura [7] has asserted that there do not exist any real hyper-surfaces in a complex projective space C

98 citations

Journal ArticleDOI
TL;DR: The idea is to use nonnegative Tucker decomposition (NTD) to obtain a set of core tensors of smaller sizes by finding a common set of projection matrices for tensor objects, and to preserve geometric information in tensor data with manifold regularization NTD.
Abstract: With the advancement of data acquisition techniques, tensor (multidimensional data) objects are increasingly accumulated and generated, for example, multichannel electroencephalographies, multiview images, and videos. In these applications, the tensor objects are usually nonnegative, since the physical signals are recorded. As the dimensionality of tensor objects is often very high, a dimension reduction technique becomes an important research topic of tensor data. From the perspective of geometry, high-dimensional objects often reside in a low-dimensional submanifold of the ambient space. In this paper, we propose a new approach to perform the dimension reduction for nonnegative tensor objects. Our idea is to use nonnegative Tucker decomposition (NTD) to obtain a set of core tensors of smaller sizes by finding a common set of projection matrices for tensor objects. To preserve geometric information in tensor data, we employ a manifold regularization term for the core tensors constructed in the Tucker decomposition. An algorithm called manifold regularization NTD (MR-NTD) is developed to solve the common projection matrices and core tensors in an alternating least squares manner. The convergence of the proposed algorithm is shown, and the computational complexity of the proposed method scales linearly with respect to the number of tensor objects and the size of the tensor objects, respectively. These theoretical results show that the proposed algorithm can be efficient. Extensive experimental results have been provided to further demonstrate the effectiveness and efficiency of the proposed MR-NTD algorithm.

98 citations

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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202326
202241
20211
20203
20192
201810