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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 article, it was shown that the kernel of a nonnegative gradient Schr\"{o}dinger operator is one dimensional provided there exists a bounded function on it and the underlying manifold is π-parabolic.
Abstract: The aim of this paper is twofold. On the one hand, the study of gradient Schr\"{o}dinger operators on manifolds with density $\phi$. We classify the space of solutions when the underlying manifold is $\phi-$parabolic. As an application, we extend the Naber-Yau Liouville Theorem, and we will prove that a complete manifold with density is $\phi -$parabolic if, and only if, it has finite $\phi-$capacity. Moreover, we show that the linear space given by the kernel of a nonnegative gradient Schr\"{o}dinger operators is one dimensional provided there exists a bounded function on it and the underlying manifold is $\phi -$parabolic. On the other hand, the topological and geometric classification of complete weighted $H_\phi -$stable hypersurfaces immersed in a manifold with density $(\amb , g, \phi)$ satisfying a lower bound on its Bakry-\'{E}mery-Ricci tensor. Also, we classify weighted stable surfaces in a three-manifold with density whose Perelman scalar curvature, in short, P-scalar curvature, satisfies $\scad + \frac{\abs{
abla \phi}^2 }{4} \geq 0$. Here, the P-scalar curvature is defined as $\scad = R - 2 \Delta _g \phi - \abs{
abla _g \phi }^2$, being $R$ the scalar curvature of $(\amb ,g)$. Finally, we discuss the relationship of manifolds with density, Mean Curvature Flow (MCF), Ricci Flow and Optimal Transportation Theory. In particular, we obtain classification results for stable self-similiar solutions to the MCF, and also for stable translating solitons to the MCF, as far as we know, this is the first classification result on stable translating solitons.
23 citations
TL;DR: In this paper, it was shown that the decomposition of the tensor product of two representations from the principal series of a Lie group consists of two pieces, Tc and Td, where Tc is a continuous direct sum with respect to Plancherel measure on G of representations from a principal series only, occurring with explicitly determined multiplicities.
Abstract: Let G be a connected semisimple real-rank one Lie group with finite center. It is shown that the decomposition of the tensor product of two representations from the principal series of G consists of two pieces, Tc and Td, where Tc is a continuous direct sum with respect to Plancherel measure on G of representations from the principal series only, occurring with explicitly determined multiplicities, and Td is a discrete sum of representations from the discrete series of G, occurring with multiplicities which are, for the present, undetermined.
23 citations
TL;DR: In this article, the contravariant components of the wave-propagation metric tensor equal half the second-order partial derivatives of the selected eigenvalue of the Christoffel matrix with respect to the slowness-vector components.
Abstract: The contravariant components of the wave-propagation metric tensor equal half the second-order partial derivatives of the selected eigenvalue of the Christoffel matrix with respect to the slowness-vector components. The relations of the wave-propagation metric tensor to the curvature matrix and Gaussian curvature of the slowness surface and to the curvature matrix and Gaussian curvature of the ray-velocity surface are demonstrated with the help of ray-centred coordinates.
23 citations
TL;DR: In this article, the authors studied complete n-dimensional Riemannian manifolds with nonnegative Ricci curvatures and large volume growth and proved that such a manifold is diffeomorphic to a Euclidean n-space if its sectional curvature is bounded from below and the volume growth of geodesic balls around some point is not too far from that of the balls in the n-dimensions.
Abstract: In this paper, we study complete open n-dimensional Riemannian manifolds with nonnegative Ricci curvature and large volume growth. We prove among other things that such a manifold is diffeomorphic to a Euclidean n-space \( R^n \) if its sectional curvature is bounded from below and the volume growth of geodesic balls around some point is not too far from that of the balls in \( R^n \).
23 citations
01 Dec 2002
TL;DR: For a positive definite fundamental tensor, all known examples of Osserman algebraic curvature tensors have a typical structure They can be produced from a metric tensor and a finite set of skew-symmetric matrices which fulfil Cliord commutation relations as mentioned in this paper.
Abstract: For a positive definite fundamental tensor all known examples of Osserman algebraic curvature tensors have a typical structure They can be produced from a metric tensor and a finite set of skew-symmetric matrices which fulfil Cliord commutation relations We show by means of Young symmetrizers and a theorem of S A Fulling, R C King, B G Wybourne and C J Cummins that every algebraic curvature tensor has a structure which is very similar to that of the above Osserman curvature tensors We verify our results by means of the Littlewood-Richardson rule and plethysms For certain symbolic calculations we used the Mathematica packages MathTensor, Ricci and PERMS
23 citations