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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 paper, it was shown that if Ricci is bounded from below, then a scalar curvature integral bound is enough to extend flow, and this integral bound condition is optimal in some sense.
Abstract: Consider {(M n , g(t)), 0 ⩽ t < T < ∞} as an unnormalized Ricci flow solution: for t ∈ [0, T). Richard Hamilton shows that if the curvature operator is uniformly bounded under the flow for all t ∈ [0, T) then the solution can be extended over T. Natasa Sesum proves that a uniform bound of Ricci tensor is enough to extend the flow. We show that if Ricci is bounded from below, then a scalar curvature integral bound is enough to extend flow, and this integral bound condition is optimal in some sense.
82 citations
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TL;DR: In this paper, it was shown that the Weyl tensor of Petrov type I can be decomposed into two parts, an electric and a magnetic part, by any observer with 4-velocity vector u. The magnetic and electric cases are distinguished by the sign of I.
Abstract: A Weyl tensor of Petrov type I can be decomposed into two parts, an electric and a magnetic part, by any observer with 4-velocity vector u. It is shown here that when a metric is such that there exists an observer who sees the metric's Weyl tensor as purely electric or purely magnetic, then the Weyl tensor is of Petrov type I in the Arianrhod--McIntosh classification (and thus its four principal null directions are linearly dependent). It is also shown that an observer exists for whom the Weyl tensor is either purely electric or magnetic if and only if the Weyl tensor is of Petrov type I and the invariant I of the Weyl tensor is real. The magnetic and electric cases are distinguished by the sign of I. In the electric and magnetic cases, the spanning vectors of the principal null directions at each point are u and two other vectors picked out by the geometry; this combines and simplifies results of Trumper and Narain. The results here are formulated in terms of invariants, and are thus easily amenable to computer classification of metrics. Spacetime examples are discussed, and new theoretical results for the Petrov type D subcase are presented.
80 citations
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TL;DR: In this paper, a general formulation of a method of reduction of Cartesian tensors, by Cartesian Tensor operations, to tensors irreducible under the three-dimensional rotation group is given.
Abstract: A general formulation is given of a method of reduction of Cartesian tensors, by Cartesian tensor operations, to tensors irreducible under the three‐dimensional rotation group. The criterion of irreducibility is that a tensor be representable as a traceless symmetric tensor, its reduced or natural form, invariantly embedded in the space of appropriate order. The general formulation exploits the properties of invariant linear mappings between tensor spaces. Considered abstractly, such mappings bring out the structure of the theory and illuminate the relation to spherical tensor theory. On the other hand, any linear invariant mapping between tensor spaces is equivalent to a combination of operations with the elementary invariant tensors U and e. The general abstract formation therefore has a direct operational representation in terms of the ordinary tensor operations of contraction and permutation of indices. An analogous formulation is given for spinors, and the relations between spinors, Cartesian tensors, and spherical tensors is discussed in the language of the present formalism. Lastly, several examples are given as to how the general formalism may be applied to groups other than the rotation group.
80 citations
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TL;DR: In this article, it was shown that a compact Kahler manifold whose Ricci tensor has two distinct constant nonnegative eigenvalues is locally the product of two Kahler-Einstein manifolds.
Abstract: It is proved that a compact Kahler manifold whose Ricci tensor has two distinct constant non-negative eigenvalues is locally the product of two Kahler–Einstein manifolds. A stronger result is established for the case of Kahler surfaces. Without the compactness assumption, irreducible Kahler manifolds with Ricci tensor having two distinct constant eigenvalues are shown to exist in various situations: there are homogeneous examples of any complex dimension n ≥ 2 with one eigenvalue negative and the other one positive or zero; there are homogeneous examples of any complex dimension n ≥ 3 with two negative eigenvalues; there are non-homogeneous examples of complex dimension 2 with one of the eigenvalues zero. The problem of existence of Kahler metrics whose Ricci tensor has two distinct constant eigenvalues is related to the celebrated (still open) conjecture of Goldberg [24]. Consequently, the irreducible homogeneous examples with negative eigenvalues give rise to complete Einstein strictly almost Kahler metrics of any even real dimension greater than 4.
80 citations
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TL;DR: In this paper, it was shown that the vanishing of the one-loop beta-functional of the doubled formalism is the same as the equation of motion of the recently proposed generalised metric formulation of double field theory restricted to this background.
80 citations