Topic
Riemann curvature tensor
About: Riemann curvature tensor is a research topic. Over the lifetime, 6248 publications have been published within this topic receiving 138871 citations. The topic is also known as: Riemann–Christoffel tensor.
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TL;DR: In this paper, the generalized flux formulation of Double Field Theory was extended to include all the first order bosonic contributions to the expansion of the heterotic string low energy effective theory, and the generalized tangent space and duality group were enhanced by $\alpha'$ corrections.
Abstract: We extend the generalized flux formulation of Double Field Theory to include all the first order bosonic contributions to the $\alpha '$ expansion of the heterotic string low energy effective theory. The generalized tangent space and duality group are enhanced by $\alpha'$ corrections, and the gauge symmetries are generated by the usual (gauged) generalized Lie derivative in the extended space. The generalized frame receives derivative corrections through the spin connection with torsion, which is incorporated as a new degree of freedom in the extended bein. We compute the generalized fluxes and find the Riemann curvature tensor with torsion as one of their components. All the four-derivative terms of the action, Bianchi identities and equations of motion are reproduced. Using this formalism, we obtain the first order $\alpha'$ corrections to the heterotic Buscher rules. The relation of our results to alternative formulations in the literature is discussed and future research directions are outlined.
86 citations
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01 Jan 1976
TL;DR: In this paper, the scalar curvature of compact riemannian manifolds was studied and the problem of finding a conformal metric for which the curvature is constant was studied.
Abstract: We shall deal with some problems concerning the scalar curvature of compact riemannian manifolds. In particular we shall deal with the problem of Yamabe: Does there exist a conformal metric for which the scalar curvature is constant? And also problems posed by Chern, Nirenberg and others. All these problems are almost entirely solved, however there remain some open questions (see the conjectures).
86 citations
18 Oct 2007
TL;DR: In this article, it was shown that if a complete Riemannian manifold supports a vector field such that the Ricci tensor plus the Lie derivative of the metric with respect to the vector field has a positive lower bound, then the fundamental group is finite.
Abstract: We show that if a complete Riemannian manifold supports a vector field such that the Ricci tensor plus the Lie derivative of the metric with respect to the vector field has a positive lower bound, then the fundamental group is finite. In particular, it follows that complete shrinking Ricci solitons and complete smooth metric measure spaces with a positive lower bound on the Bakry-Emery tensor have finite fundamental group. The method of proof is to generalize arguments of Garcia-Rio and Fernandez-Lopez in the compact case.
86 citations
TL;DR: The geometrical form of the one-loop divergences induced by conical singularities of background manifolds is studied in this article, where surface corrections to standard second and third heat coefficients are obtained explicitly in terms of angle α of a cone Cα and components of the Riemann tensor.
86 citations
TL;DR: A framework for decomposing variations in diffusion tensors into changes in shape and orientation is introduced, including a direct approximation for the variance of tensor invariants such as fractional anisotropy.
Abstract: Guided by empirically established connections between clinically important tissue properties and diffusion tensor parameters, we introduce a framework for decomposing variations in diffusion tensors into changes in shape and orientation. Tensor shape and orientation both have three degrees-of-freedom, spanned by invariant gradients and rotation tangents, respectively. As an initial demonstration of the framework, we create a tunable measure of tensor difference that can selectively respond to shape and orientation. Second, to analyze the spatial gradient in a tensor volume (a third-order tensor), our framework generates edge strength measures that can discriminate between different neuroanatomical boundaries, as well as creating a novel detector of white matter tracts that are adjacent yet distinctly oriented. Finally, we apply the framework to decompose the fourth-order diffusion covariance tensor into individual and aggregate measures of shape and orientation covariance, including a direct approximation for the variance of tensor invariants such as fractional anisotropy.
86 citations