Topic
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 article, the authors show that the SA-decomposition is unique, irreducible, and preserves the symmetries of the elasticity tensor in linear anisotropic elasticity.
Abstract: In linear anisotropic elasticity, the elastic properties of a medium are described by the fourth rank elasticity tensor C. The decomposition of C into a partially symmetric tensor M and a partially antisymmetric tensors N is often used in the literature. An alternative, less well-known decomposition, into the completely symmetric part S of C plus the reminder A, turns out to be irreducible under the 3-dimensional general linear group. We show that the SA-decomposition is unique, irreducible, and preserves the symmetries of the elasticity tensor. The MN-decomposition fails to have these desirable properties and is such inferior from a physical point of view. Various applications of the SA-decomposition are discussed: the Cauchy relations (vanishing of A), the non-existence of elastic null Lagrangians, the decomposition of the elastic energy and of the acoustic wave propagation. The acoustic or Christoffel tensor is split in a Cauchy and a non-Cauchy part. The Cauchy part governs the longitudinal wave propagation. We provide explicit examples of the effectiveness of the SA-decomposition. A complete class of anisotropic media is proposed that allows pure polarizations in arbitrary directions, similarly as in an isotropic medium.
21 citations
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TL;DR: In this article, the authors showed that the space of potential tensor fields is a subspace of a finite codimension in Ker I if M is simple, i.e., every two points are joined by a unique geodesic.
Abstract: The ray transform I on a compact Riemannian manifold M with boundary is the operator sending a symmetric tensor field f to the set of integrals of f over all geodesics joining boundary points. A field f is called potential if it can be represented as the symmetric part of the covariant derivative of another tensor field vanishing on the boundary: The main result asserts that the space of potential tensor fields is a subspace of a finite codimension in Ker I if M is simple. A Riemannian manifold is called simple if every two points are joined by a unique geodesic.
21 citations
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01 Jan 1981
TL;DR: In this article, the Ricci tensor of a compact Riemannian manifold with harmonic curvature must be parallel, and Bourguignon has shown that it is not the case.
Abstract: While every manifold with parallel Ricci tensor has harmonic curvature (i.e. satisfies 6R = 0), there are examples ([3], Theorem 5.2) of open Riemannian manifolds with 6R = 0 and VS ~ O. In [1] J.P. Bourguignon has asked the question whether the Ricci tensor of a compact Riemannian manifold with harmonic curvature must be parallel. The aim of this note is to describe an easy example answering this question in the negative. More precisely, metrics with 6R = 0 and VS ~ 0 are exhibited on S 1 × N 3, N 3 being e.g. the 3-sphere or a lens space. By taking products of these manifolds with themselves or with arbitrary compact Einstein manifolds, one gets similar examples in all dimensions greater {han three.
21 citations
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TL;DR: In this article, the authors describe results on harmonic functions of poly-nomial growth on open manifolds with nonnegative Ricci curvature and Euclidean volume growth, and show that these functions can be computed in polynomial time.
Abstract: In this announcement, we describe some results of an ongoing investigation of function theoryon spaces with a lower Ricci curvature bound. In particular, we announce results on harmonic functions of poly- nomial growth on open manifolds with nonnegative Ricci curvature and Euclidean volume growth.
21 citations
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21 citations