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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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Journal ArticleDOI
Naoki Sasakura1
TL;DR: In this paper, the relation between rank-three tensor models and the dynamical triangulation model of three-dimensional quantum gravity is investigated, and the orientability of the manifold and the corresponding tensors models is discussed.
Abstract: We investigate the relation between rank-three tensor models and the dynamical triangulation model of three-dimensional quantum gravity, and discuss the orientability of the manifold and the corresponding tensor models. We generalize the orientable tensor models to arbitrary dimensions, which include the two-dimensional Hermitian matrix model as a special case.

301 citations

Journal ArticleDOI
TL;DR: In this paper, a self-contained presentation of this formalism is given, including a discussion of the constraints and its solutions, and of the resulting Riemann tensor, Ricci tensor and curvature scalar.
Abstract: We relate two formulations of the recently constructed double field theory to a frame-like geometrical formalism developed by Siegel. A self-contained presentation of this formalism is given, including a discussion of the constraints and its solutions, and of the resulting Riemann tensor, Ricci tensor and curvature scalar. This curvature scalar can be used to define an action, and it is shown that this action is equivalent to that of double field theory.

298 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that the Ricci curvature of a Riemannian metric is either everywhere positive or everywhere negative, and the divergence and curl both vanish.
Abstract: We shall prove theorems on nonexistence of certain types of vector fields on a compact manifold with a positive definite Riemannian metric whose Ricci curvature is either everywhere positive or everywhere negative. Actually we shall have some relaxations of the requirements both as to curvature and as to compactness. We shall deal with real spaces with a customary metric and with complex analytic spaces with an Hermitian metric. In the latter case we shall impose on the metric a certain restriction, first explicitly stated by E. Kaehler, which will be quite indispensable to our argument. In order to elucidate the rôle of this restriction we shall include a systematic introduction to the theory of Hermitian metric. For positive curvature we shall have the theorem that on a compact space there exists no vector field for which the divergence and curl both vanish. In the complex case there exists no vector field whatsoever whose covariant components are analytic functions in the complex parameters. If we only assume that the curvature is nonnegative, then there are some \"exceptional\" vector fields in directions of spatial flatness. A principal result will be the following theorem on meromorphic functions. If a complex space with positive curvature is covered by a finite number of neighborhoods, if a meromorphic functional element is defined in each neighborhood, and if the difference of any two meromorphic elements is holomorphic wherever the elements overlap, then there exists one meromorphic function on the space which differs by a holomorphic function from each meromorphic element given. In a previous paper this conclusion was drawn in the

296 citations

Journal ArticleDOI
TL;DR: The theoretical developments provide a method for generating scalar maps of the diffusion tensor data, including novel fractional anisotropy maps that are color encoded for the mode of anisOTropy and directionally encoded colormaps of only linearly anisotropic structures, rather than of high fractionalAnisotropic structures.
Abstract: This paper outlines the mathematical development and application of two analytically orthogonal tensor invariants sets. Diffusion tensors can be mathematically decomposed into shape and orientation information, determined by the eigenvalues and eigenvectors, respectively. The developments herein orthogonally decompose the tensor shape using a set of three orthogonal invariants that characterize the magnitude of isotropy, the magnitude of anisotropy, and the mode of anisotropy. The mode of anisotropy is useful for resolving whether a region of anisotropy is linear anisotropic, orthotropic, or planar anisotropic. Both tensor trace and fractional anisotropy are members of an orthogonal invariant set, but they do not belong to the same set. It is proven that tensor trace and fractional anisotropy are not mutually orthogonal measures of the diffusive process. The results are applied to the analysis and visualization of diffusion tensor magnetic resonance images of the brain in a healthy volunteer. The theoretical developments provide a method for generating scalar maps of the diffusion tensor data, including novel fractional anisotropy maps that are color encoded for the mode of anisotropy and directionally encoded colormaps of only linearly anisotropic structures, rather than of high fractional anisotropy structures.

269 citations

Journal ArticleDOI
TL;DR: In this article, the question of Lipschitz convergence of compact Riemannian manifolds with bounds imposed on the Ricci curvature Ric in was studied. And it was shown that Gromov's compactness theorem may be strengthened to the statement that f((A, v, D) is C1 'l compact in the Lipschnitz topology.
Abstract: where f: MO M1 is a homeomorphism and dil f is the dilatation of f given by dil f = supX,X2 dist(f(x) , f(x2))/ dist(x1, x2) . If MO and M1 are not homeomorphic, define dL(MO, MI) = +oo. Gromov [20] proves the remarkable result that the space of compact Riemannian manifolds f((A, 3, D) of sectional curvature IKI 3 > 0, and diameter dM v, and diameter dM c(IKI , dm, VM1) In particular, Gromov's compactness theorem may be strengthened to the statement that f((A, v , D) is C1 'l compact in the Lipschitz topology. In this paper, we study the question of Lipschitz convergence of compact Riemannian manifolds with bounds imposed on the Ricci curvature Ric in

267 citations

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