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.

Primordial non-Gaussianities in general modified gravitational models of inflation

Abstract: We compute the three-point correlation function of primordial scalar density perturbations in a general single-field inflationary scenario, where a scalar field has a direct coupling with the Ricci scalar R and the Gauss-Bonnet term . Our analysis also covers the models in which the Lagrangian includes a function non-linear in the field kinetic energy X = −(∂)2/2, and a Galileon-type field self-interaction G(,X), where G is a function of and X. We provide a general analytic formula for the equilateral non-Gaussianity parameter fNLequil associated with the bispectrum of curvature perturbations. A quasi de Sitter approximation in terms of slow-variation parameters allows us to derive a simplified form of fNLequil convenient to constrain various inflation models observationally. If the propagation speed of the scalar perturbations is much smaller than the speed of light, the Gauss-Bonnet term as well as the Galileon-type field self-interaction can give rise to large non-Gaussianities testable in future observations. We also show that, in Brans-Dicke theory with a field potential (including f(R) gravity), fNLequil is of the order of slow-roll parameters as in standard inflation driven by a minimally coupled scalar field.

Riemannian Geometry During the Second Half of the Twentieth Century

Abstract: Additional bibliography Riemannian geometry up to 1950 Comments on the main topics I, II, III, IV, V under consideration Curvature and topology The geometrical hierarchy of Riemann manifolds: Space forms The set of Riemannian structures on a given compact manifold: Is there a best metric? The spectrum, the eigenfunctions Periodic geodesics, the geodesic flow Some other Riemannian geometric topics of interest Bibliography Subject and notation index Name index

Curvature Functions for Open 2-Manifolds

Abstract: The basic problem posed in [12] is that of describing the set of Gaussian curvature functions which a given 2-dimensional manifold M can possess. In this paper we consider this problem for the case of non-compact M. Other than the Gauss-Bonnet type inequality of Cohn-Vossen [4] (see also [6], [8]), which holds for certain complete metrics on non-compact manifolds, there is no known a priori restriction for a Gaussian curvature function on a non-compact 2-manifold. Indeed, Gromov has recently shown that any non-compact 2-manifold possesses a metric of strictly positive curvature as well as a metric of strictly negative curvature [7]. As the analogue of Question 1 of [12] it therefore seems natural to ask the following:

Projectively flat Finsler metrics of constant flag curvature

Abstract: Finsler metrics on an open subset in R n with straight geodesics are said to be projective. It is known that the flag curvature of any projective Finsler metric is a scalar function of tangent vectors (the flag curvature must be a constant if it is Riemannian). In this paper, we discuss the classification problem on projective Finsler metrics of constant flag curvature. We express them by a Taylor expansion or an algebraic formula. Many examples constructed in this paper can be used as models in Finsler geometry.

Bach-flat asymptotically locally Euclidean metrics

Abstract: We obtain a volume growth and curvature decay result for various classes of complete, noncompact Riemannian metrics in dimension 4; in particular our method applies to anti-self-dual or Kahler metrics with zero scalar curvature, and metrics with harmonic curvature. Similar results were obtained for Einstein metrics in [And89], [BKN89], [Tia90], but our analysis differs from the Einstein case in that (1) we consider more generally a fourth order system in the metric, and (2) we do not assume any pointwise Ricci curvature bound.