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 paper, a non-flat Riemannian manifold called pseudo cyclic Ricci symmetric manifold was introduced and its geometric properties were studied. And it was shown that such a manifold can be isometrically immersed in a Euclidean manifold as a hypersurface.
Abstract: The object of the present paper is to introduce a type of non-flat Riemannian manifold called pseudo cyclic Ricci symmetric manifold and study its geometric properties. Among others it is shown that a pseudo cyclic Ricci symmetric manifold is a special type of quasi-Einstein manifold. In this paper we also study conformally flat pseudo cyclic Ricci symmetric manifolds and prove that such a manifold can be isometrically immersed in a Euclidean manifold as a hypersurface.
8 citations
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TL;DR: In this article, the authors studied holomorphically planar conformal vector fields (HPCV) on contact metric manifolds under some curvature conditions, and they showed that HPCV fields on these manifolds are locally isometric to En+1×Sn(4).
Abstract: We study holomorphically planar conformal vector fields (HPCV) on contact metric manifolds under some curvature conditions. In particular, we have studied HPCV fields on (i) contact metric manifolds with pointwise constant ξ-sectional curvature (under this condition M is either K-contact or V is homothetic), (ii) Einstein contact metric manifolds (in this case M becomes K contact), (iii) contact metric manifolds with parallel Ricci tensor (under this condition M is either K-contact Einstein or is locally isometric to En+1×Sn(4)).
8 citations
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TL;DR: In this article, the Ricci curvature of a Riemannian manifold is bounded from below, and the geometry of the manifold at infinity is not too extreme, and it is shown that a Ricci flow exists for a short time interval.
Abstract: We consider complete (possibly non-compact) three dimensional Riemannian manifolds (M,g) such that: a) (M,g) is non-collapsed, b) the Ricci curvature of (M,g) is bounded from below, c) the geometry of (M,g) at infinity is not too extreme. Given such initial data (M,g) we show that a Ricci flow exists for a short time interval. This enables us to construct a Ricci flow of any (possibly singular) metric space (X,d) which arises as a Gromov-Hausdorff limit of a sequence of 3-manifolds which satisfy a), b) and c) uniformly. As a corollary we show that such an X must be a manifold.
8 citations
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TL;DR: In this paper, the curvature conditions on manifolds admitting a symmetry group are studied, including the extremal curvature condition on Kahler toric manifolds, and the Stenzel invariant on the Fubini-Study metric on complex projective spaces.
Abstract: Various curvature conditions are studied on metrics admitting a symmetry group. We begin by examining a method of diagonalizing cohomogeneity-one Einstein manifolds and determine when this method can and cannot be used. Examples, including the well-known Stenzel metrics, are discussed. Next, we present a simplification of the Einstein condition on a compact four manifold with $T^{2}$-isometry to a system of second-order elliptic equations in two-variables with well-defined boundary conditions. We then study the Einstein and extremal Kahler conditions on Kahler toric manifolds. After constructing explicitly new extremal Kahler and constant scalar curvature metrics, we demonstrate how these metrics can be obtained by continuously deforming the Fubini-Study metric on complex projective space in dimension three. We also define a generalization of Kahler toric manifolds, which we call fiberwise Kahler toric manifolds, and construct new explicit extremal Kahler and constant scalar curvature metrics on both compact and non-compact manifolds in all even dimensions. We also calculate the Futaki invariant on manifolds of this type. After describing an Hermitian non-Kahler analogue to fiberwise Kahler toric geometry, we construct constant scalar curvature Hermitian metrics with $J$-invariant Riemannian tensor. In dimension four, we write down explicitly new constant scalar curvature Hermitian metrics with $J$-invariant Ricci tensor. Finally, we integrate the scalar curvature equation on a large class of cohomogeneity-one metrics.
8 citations
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TL;DR: In this paper, the classical mechanics associated with a conformally flat Riemannian metric on a compact, n-dimensional manifold without boundary were constructed and the corresponding gradient Ricci flow equation was shown to equal the time-dependent Hamilton-Jacobi equation.
Abstract: We construct the classical mechanics associated with a conformally flat Riemannian metric on a compact, n-dimensional manifold without boundary. The corresponding gradient Ricci flow equation turns out to equal the time-dependent Hamilton–Jacobi equation of the mechanics so defined.
8 citations