Ricci decomposition

About: Ricci decomposition is a research topic. Over the lifetime, 1972 publications have been published within this topic receiving 45295 citations.

Nonassociative differential geometry and gravity with non-geometric fluxes

Abstract: We systematically develop the metric aspects of nonassociative differential geometry tailored to the parabolic phase space model of constant locally non-geometric closed string vacua, and use it to construct preliminary steps towards a nonassociative theory of gravity on spacetime. We obtain explicit expressions for the torsion, curvature, Ricci tensor and Levi-Civita connection in nonassociative Riemannian geometry on phase space, and write down Einstein field equations. We apply this formalism to construct R-flux corrections to the Ricci tensor on spacetime, and comment on the potential implications of these structures in non-geometric string theory and double field theory.

Comment: Towards a Complete Classification of Spherically Symmetric Lorentzian Manifolds According to Their Ricci Collineations

Abstract: General expressions for the components of the Ricci collineation vector are derived and the related constraints are obtained. These constraints are then solved to obtain Ricci collineations and the related constraints on the Ricci tensor components for all spacetime manifolds (degenerate or non-degenerate, diagonal or non-diagonal) admitting symmetries larger than so(3) and already known results are recovered. A complete solution is achieved for the spacetime manifolds admitting so(3) as the maximal symmetry group with non-degenerate and non diagonal Ricci tensor components. It is interesting to point out that there appear cases with finite number of Ricci collineations although the Ricci tensor is degenerate and also the cases with infinitely many Ricci collineations even in the case of non-degenerate Ricci tensor. Interestingly, it is found that the spacetime manifolds with so(3) as maximal symmetry group may admit two extra proper Ricci collineations, although they do not admit a G5 as the maximal symmetry group. Examples are provided which show and clarify some comments made by Camci et al. [Camci, U., and Branes, A. (2002). Class. Quantum Grav.19, 393–404]. Theorems are proved which correct the earlier claims made in [Carot, J., Nunez, L. A., and Percoco, U. (1997). Gen. Relativ. Gravit.29, 1223–1237; Contreras, G., Nunez, L. A., and Percolo, U. (2000). Gen. Relativ. Gravit.32, 285–294].

Tensor products of spherical and equivariant immersions

Abstract: In [C1, C2, C3, C4], B.-Y. Chen introduced the tensor product of two immersions of a given Riemannian manifold; he proved that the set of all immersions of the given manifold, provided with direct sum and tensor product, defines a commutative semiring. In [DDVV] we introduced I, the commutative semiring of all transversal immersions of all differentiable manifolds in Euclidean spaces, provided with the binary operations direct sum and tensor product. In this paper we further investigate which immersions define a subsemiring or a multiplicative subsemigroup ; in particular, we fix our attention on spherical immersions of differentiable manifolds, isometric and equivariant immersions of Riemannian manifolds and immersions of finite type. Denote by E the n-dimensional Euclidean space with Euclidean metric 〈 , 〉. The n-dimensional sphere with radius r is denoted by S(r). Let f : M → E be an immersion of a differentiable manifold in a Euclidean space. Then f is said to be transversal in a point p ∈M if and only if the position vector f(p) is not tangent to M at p, i.e. f(p) / ∈ f∗(TpM). If f is transversal in every point of M , then f shortly is called transversal. Consider two differentiable manifolds M and N of dimensions r resp. s and assume that f : M → E and h : N → E are two transversal immersions. Then the direct sum map f⊕h : M×N → E : (p, q) 7→ (f(p), h(q)) and the tensor product map f ⊗ h : M × N → E : (p, q) 7→ f(p) ⊗ h(q) are again two transversal immersions. We define a symmetric relation ∼ as follows : if f : M → E is an immersion and i : E ⊂ E is a linear isometric immersion, then ∗Supported by a research fellowship of the Research Council of the Katholieke Universiteit Leuven †Senior Research Assistant of the National Fund for Scientific Research (Belgium) ‡Research Fellow of the Research Council of the K.U.Leuven Received by the editors November 1993 Communicated by A. Warrinier AMS Mathematics Subject Classification : 53C40, 53B25, 58G25