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, it was shown that all (2j-1)-connected 2j-parallelisable manifolds in dimension 4j+1, j > 0, can be equipped with Riemannian metrics of 2-positive Ricci curvature.
Abstract: We show that after forming a connected sum with a homotopy sphere, all (2j-1)-connected 2j-parallelisable manifolds in dimension 4j+1, j > 0, can be equipped with Riemannian metrics of 2-positive Ricci curvature. The condition of 2-positive Ricci curvature is defined to mean that the sum of the two smallest eigenvalues of the Ricci tensor is positive at every point. This result is a counterpart to a previous result of the authors concerning the existence of positive Ricci curvature on highly connected manifolds in dimensions 4j-1 for j > 1, and in dimensions 4j+1 for j > 0 with torsion-free cohomology.
13 citations
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TL;DR: In this paper, a generalized quasi-conformal curvature tensor (GQC tensor) was introduced, which is a new curva-ture tensor which bridges conformal curvatures, concircular curvatures and conharmonic curvatures.
Abstract: . The object of the present paper is to introduce a new curva-ture tensor, named generalized quasi-conformal curvature tensor whichbridges conformal curvature tensor, concircular curvature tensor, pro-jective curvature tensor and conharmonic curvature tensor. Flatness andsymmetric properties of generalized quasi-conformal curvature tensor arestudied in the frame of (k,µ)-contact metric manifolds. 1. IntroductionIn 1968, Yano and Sawaki [27] introduced the notion of quasi-conformalcurvature tensor which contains both conformal curvature tensor as well asconcircular curvature tensor, in the context of Riemannian geometry. In tunewith Yano and Sawaki [27], the present paper attempts to introduce a newtensor field, named generalized quasi-conformal curvature tensor. The beautyof generalized quasi-conformal curvature tensor lies in the fact that it has theflavour of Riemann curvature tensor R, conformal curvature tensor C [8] con-harmonic curvature tensor Cˆ [9], concircular curvature tensor E [26, p. 84],projective curvature tensor P [26, p. 84] and m-projective curvature tensor H[15], as particular cases. The generalized quasi-conformal curvature tensor isdefined asW(X,Y)Z =2n−12n+1[(1−b+2na)−{1+2n(a+b)}c]C(X,Y )Z+[1−b+2na]E(X,Y)Z +2 n (b−a) P(X,Y )Z+2 n−12 n+1(1.1) (c −1){1+2 n(a +b)} Cˆ(X,Y)Zfor all X,Y,Z ∈ χ(M), the set of all vector field of the manifold M, where a,b and c are real constants. The above mentioned curvature tensors are defined
13 citations
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13 citations
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TL;DR: In this article, the Bel-Robinson tensor is analyzed as a linear map on the space of the traceless symmetric tensors and an algebraic classification that refines the usual Petrov-bel classification of the Weyl tensors is presented.
Abstract: The Bel–Robinson tensor is analyzed as a linear map on the space of the traceless symmetric tensors. This study leads to an algebraic classification that refines the usual Petrov–Bel classification of the Weyl tensor. The new classes correspond to degenerate type I space-times which have already been introduced in literature from another point of view. The Petrov–Bel types and the additional ones are intrinsically characterized in terms of the sole Bel–Robinson tensor, and an algorithm is proposed that enables the different classes to be distinguished. Results are presented that solve the problem of obtaining the Weyl tensor from the Bel–Robinson tensor in regular cases.
13 citations