Topic
Scalar curvature
About: Scalar curvature is a research topic. Over the lifetime, 12701 publications have been published within this topic receiving 296040 citations. The topic is also known as: Ricci scalar.
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TL;DR: In this paper, it was shown that the model of Vollick does not have a good Newtonian limit, and also that any theory with a pole of order n in $R=0$ and a constant curvature term, where R is the scalar curvature of background, has no good limit.
Abstract: Recently D. Vollick [Phys. Rev. D 68, 063510 (2003)] has shown that the inclusion of the $1/R$ curvature terms in the gravitational action and the use of the Palatini formalism offer an alternative explanation for cosmological acceleration. In this work we show not only that this model of Vollick does not have a good Newtonian limit, but also that any $f(R)$ theory with a pole of order n in $R=0$ and ${(d}^{2}{f/d}^{2}{R)(R}_{0})\ensuremath{
e}0,$ where ${R}_{0}$ is the scalar curvature of background, does not have a good Newtonian limit.
95 citations
TL;DR: In this article, a description of all invariant Riemannian metrics on manifolds of sectorial states is given, and the equations of the geodesies for the entire family of invariant linear connections Δ=γΔ, γ∈ IR are integrated on sets of classical probability distributions.
Abstract: This paper is devoted to certain differential-geometric constructions in classical and noncommutative statistics, invariant with respect to the category of Markov maps, which have recently been developed by Soviet, Japanese, and Danish researchers. Among the topics considered are invariant metrics and invariant characteristics of informational proximity, and lower bounds are found for the uniform topologies that they generate on sets of states. A description is given of all invariant Riemannian metrics on manifolds of sectorial states. The equations of the geodesies for the entire family of invariant linear connections Δ=γΔ, γ∈ IR, are integrated on sets of classical probability distributions. A description is given of the protective structure of all the geodesic curves and totally geodesic submanifolds, which turns out to be a local lattice structure; it is shown to coincide, up to a factor γ(γ−1), with the Riemann-Christoffel curvature tensor.
95 citations
95 citations
TL;DR: In this article, the scalar curvature of 7-dimensional manifolds admitting a connection with totally skew-symmetric torsion is computed in terms of the dilation function and the NS 3-form field.
Abstract: We compute the scalar curvature of 7-dimensional ${G}_2$-manifolds admitting a connection with totally skew-symmetric torsion. We prove the formula for the general solution of the Killing spinor equation and express the Riemannian scalar curvature of the solution in terms of the dilation function and the NS 3-form field. In dimension n=7 the dilation function involved in the second fermionic string equation has an interpretation as a conformal change of the underlying integrable ${G}_2$-structure into a cocalibrated one of pure type $W_3$.
94 citations