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.

Proof of summed form of proper-time expansion for propagator in curved space-time.

Abstract: We consider the Schwinger-DeWitt proper-time expansion of the kernel of the Feynman propagator in curved space-time. We prove that the proper-time expansion can be written in a new form, conjectured by Parker and Toms, in which all the terms containing the scalar curvature R are generated by a simple overall exponential factor. This sums all terms containing R, including those with nonconstant coefficients, in the proper-time series. This result is valid for an arbitrary space-time and for any spin. It also applies to the heat kernel. This form of the expansion is of importance in connection with nonperturbative effects in quantum field theory.

Metric Inequalities with Scalar Curvature

Abstract: We establish several inequalities for manifolds with positive scalar curvature and, more generally, for the scalar curvature bounded from below. In so far as geometry is concerned these inequalities appear as generalisations of the classical bounds on the distances between conjugates points in surfaces with positive sectional curvatures. The techniques of our proofs is based on the Schoen–Yau descent method via minimal hypersurfaces, while the overall logic of our arguments is inspired by and closely related to the torus splitting argument in Novikov’s proof of the topological invariance of the rational Pontryagin classes.

On the stability of Riemannian manifold with parallel spinors

Abstract: Inspired by the recent work [HHM03], we prove two stability results for compact Riemannian manifolds with nonzero parallel spinors. Our first result says that Ricci flat metrics which also admit nonzero parallel spinors are stable (in the direction of changes in conformal structures) as the critical points of the total scalar curvature functional. Our second result, which is a local version of the first one, shows that any metric of positive scalar curvature cannot lie too close to a metric with nonzero parallel spinor. We also prove a rigidity result for special holonomy metrics. In the case of SU(m) holonomy, the rigidity result implies that scalar flat deformations of Calabi-Yau metric must be Calabi-Yau. Finally we explore the connection with a positive mass theorem of [D03], which presents another approach to proving these stability and rigidity results.

An elementary method in the study of nonnegative curvature

Abstract: A standard technique in classical analysis for the study of eontinous sub-solutions of the Dirichlet problem for second order operators may be illustrated as follows. Suppose it is to be shown that a continuous real function ](x) is convex (respectively, striely convex) at x0; then it suffices to produce a C ~ function g(x) such that g(x)<<.](x) near x 0 and g(Xo) =/(x0), and such that 9\"(xo) >/0 (respectively g\"(xo) >1 some fixed positive constant). The main point of this procedure is to sidestep arguments involving continuous functions by working with differentiable functions alone. Now in global differential geometry, the functions that naturally arise are often continuous but not differentiable. Since much of geometric analysis reduces to second order elliptic problems, this technique then recommends itself as a natural tool for overcoming this difficulty with the lack of differentiability. In a limited way, this technique has indeed appeared in several papers in complex geometry (e.g. Ahlfors [1], Takeuchi [20], Elenewajg [7] and Greene-Wu [11]; cf. also Suzuki [19]). The main purpose of this paper is to broaden and deepen the scope of this method by making it the central point of a general study of nonnegative sectional, Ricei or bisectional curvature. The following are the principal theorems; the relevant definitions can be found in Section 1. Let M be a noncompact complete Riemannian manifold and let 0 E M be fixed. Let {Ct}tG1 be a family of closed subsets of M indexed by a subset I of R. Assume that et = d(0, C t ) ~ as t ~ , where d(p, q) will always denote the distance between p, qEM relative to the Riemannian metric. The family of functions ~t: M-~R defined by ~t(P)=

Classification of manifolds with weakly 1/4-pinched curvatures

Abstract: We show that a compact Riemannian manifold with weakly pointwise 1/4-pinched sectional curvatures is either locally symmetric or diffeomorphic to a space form. More generally, we classify all compact, locally irreducible Riemannian manifolds M with the property that M × R 2 has non-negative isotropic curvature.