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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 a compact 7-dimensional manifold equipped with a G2-structure with closed fundamental form is Einstein if and only if the Riemannian holonomy of the induced metric is contained in G2.
Abstract: We give an answer to a question posed in physics by Cvetic et al [9] and recently in mathematics by Bryant [3], namely we show that a compact 7-dimensional manifold equipped with a G2-structure with closed fundamental form is Einstein if and only if the Riemannian holonomy of the induced metric is contained in G2 This could be considered to be a G2 analogue of the Goldberg conjecture in almost Kahler geometry and was indicated by Cvetic et al in [9] The result was generalized by Bryant to closed G2-structures with too tightly pinched Ricci tensor We extend it in another direction proving that a compact G2-manifold with closed fundamental form and divergence-free Weyl tensor is a G2-manifold with parallel fundamental form We introduce a second symmetric Ricci-type tensor and show that Einstein conditions applied to the two Ricci tensors on a closed G2-structure again imply that the induced metric has holonomy group contained in G2
65 citations
TL;DR: In this article, the authors introduced the notion of parallel Ricci tensor for real hypersurfaces in the complex quadric Qm=SOm+2/SOmSO2.
64 citations
TL;DR: Tangent space null rotations are used to give a straightforward classification of the Ricci tensor in general relativity theory as discussed by the authors, and they are used for the classification of Ricci Tensor.
Abstract: Tangent space null rotations are used to give a straightforward classification of the Ricci tensor in general relativity theory.
64 citations
64 citations
TL;DR: In this paper, the problem of finding physically pertinent stationary axially symmetric Einstein-Maxwell fields is reduced to the search for a complex solution ξ0(x, y) of one nonlinear differential equation subject to simple subsidiary conditions.
Abstract: Spin‐coefficients and null tetrad components of the Ricci tensor and the Weyl conform tensor are evaluated in terms of a single complex gravitational potential e, while null tetrad components of the electromagnetic stress energy tensor are evaluated in terms of a second complex potential φ. All the results are expressed elegantly in terms of a differential operator ð, similar to the ``thop'' of Newman and Penrose. The problem of finding physically pertinent stationary axially symmetric Einstein‐Maxwell fields is reduced to the search for a complex solution ξ0(x, y) of one nonlinear differential equation subject to simple subsidiary conditions.
63 citations