scispace - formally typeset
Search or ask a question
Topic

Ricci decomposition

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


Papers
More filters
Journal ArticleDOI
TL;DR: The analysis of the admissibility of a potential representation for the Riemann tensor is continued in this article, where it is shown that there never exist ordinary solutions in a four-dimensional manifold and the existence of singular solutions is established without requiring any integrability condition.
Abstract: The analysis of the admissibility of a potential representation for the Riemann tensor is here continued. As in the preceding paper, the starting point is to regard the relationship between the Riemann tensor and its possible potential as a system of partial differential equations determining the unknown potential. The first result, strengthening a previous conclusion, is that there never exist ordinary solutions. Surprisingly enough, in a four-dimensional Riemannian manifold the existence of singular solutions is established without requiring any integrability condition. Possible applications and generalizations are also suggested.

9 citations

Journal ArticleDOI
TL;DR: In this paper, the difference between the third-order tensor potential and the Lanczos generating function of the Weyl tensor is characterized by a vector obtained by contraction, and the significant role of such a vector, in the context of general relativity, is discussed.
Abstract: In a Riemannian space-time, the difference between the third-order tensor potentialH αβλ of the Riemann tensor (presented in a precedent paper) and the Lanczos generating function of the Weyl tensor is here shown to be characterized by a vectorV α , obtained by contractionH αβλ . The significant role of such a vector, in the context of general relativity, is then discussed. Particular attention is paid to the scalar potential ϑ which characterizes the irrotational part ofV α : such a scalar field satisfies a space-time wave equation of the Poisson type. Weak fields are also considered: in the particular case of a static metric, the scalar ϑ is found to be proportional to the classic Newtonian potential.

9 citations

Journal ArticleDOI
TL;DR: In this paper, the Ricci tensor of Walker manifolds of signature (2, 2) is shown with various commutativity properties for Ricci operator, skew-symmetric curvature operator and Jacobi operator.
Abstract: We exhibit Walker manifolds of signature (2, 2) with various commutativity properties for the Ricci operator, the skew-symmetric curvature operator and the Jacobi operator. If the Walker metric is a Riemannian extension of an underlying affine structure , these properties are related to the Ricci tensor of .

9 citations

Journal ArticleDOI
TL;DR: In this article, the authors review, establish, and compare the perturbation bounds for two natural types of incremental rank-one approximation approaches, and present numerical experiments and open questions.
Abstract: Finding the symmetric and orthogonal decomposition of a tensor is a recurring problem in signal processing, machine learning, and statistics. In this paper, we review, establish, and compare the perturbation bounds for two natural types of incremental rank-one approximation approaches. Numerical experiments and open questions are also presented and discussed.

9 citations

Proceedings Article
01 Jan 2005
TL;DR: In this paper, exact quadrature formulae for mean curvature, Gaussian curvature and the Taubin integral representation of the curvature tensor are derived for a smooth surface approximated by a dense triangle mesh.
Abstract: Accurate estimations of geometric properties of a surface from its discrete approximation are important for many computer graphics and geometric modeling applications. In this paper, we derive exact quadrature formulae for mean curvature, Gaussian curvature, and the Taubin integral representation of the curvature tensor. The exact quadratures are then used to obtain reliable estimates of the curvature tensor of a smooth surface approximated by a dense triangle mesh. The proposed method is fast and easy to implement. It is highly competitive with conventional curvature tensor estimation approaches. Additionally, we show that the curvature tensor approximated as proposed by us converges towards the true curvature tensor as the edge lengths tend to zero.

9 citations

Network Information
Related Topics (5)
Lie group
18.3K papers, 381K citations
85% related
Operator theory
18.2K papers, 441.4K citations
84% related
Cohomology
21.5K papers, 389.8K citations
82% related
Abelian group
30.1K papers, 409.4K citations
81% related
Space (mathematics)
43K papers, 572.7K citations
81% related
Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202326
202241
20211
20203
20192
201810