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.

Rigidity of Einstein 4-manifolds with positive curvature

Abstract: An Einstein metric with positive scalar curvature on a 4-manifold is said to be normalized if Ric=1. A basic problem in Riemannian geometry is to classify Einstein 4-manifolds with positive sectional curvature in the category of either topology, diffeomorphism, or isometry. It is shown in this paper that if the sectional curvature K of a normalized Einstein 4-manifold M satisfies the lower bound K≥e0, e0≡( -23)/120≈0.102843, or condition (b) of Theorem 1.1, then it is isometric to either S 4, RP 4 with constant sectional curvature K=1/3, or CP 2 with the normalized Fubini-Study metric. As a consequence, both the normalized moduli spaces of Einstein metrics which satisfy either one of the above two conditions on S 4 and CP 2 contain only a single point. In particular, if M is a smooth 4-manifold which is homeomorphic to either S 4, RP 4, or CP 2 but not diffeomorphic to any of the three manifolds, then it can not support any normalized Einstein metric which satisfies either one of the conditions.

Comparison and rigidity theorems in semi-Riemannian geometry

Abstract: The comparison theory for the Riccati equation satis- ed by the shape operator of parallel hypersurfaces is generalized to semi{Riemannian manifolds of arbitrary index, using one{sided bounds on the Riemann tensor which in the Riemannian case cor- respond to one{sided bounds on the sectional curvatures. Starting from 2{dimensional rigidity results and using an inductive tech- nique, a new class of gap{type rigidity theorems is proved for semi{ Riemannian manifolds of arbitrary index, generalizing those rst given by Gromov and Greene{Wu. As applications we prove rigid- ity results for semi{Riemannian manifolds with simply connected ends of constant curvature.

Optimal Sobolev inequalities on complete Riemannian manifolds with Ricci curvature bounded below and positive injectivity radius

Abstract: The concept of best constants for Sobolev embeddings appeared to be crucial for solving limiting cases of some partial differential equations. A striking example where it has played a major role is given by the very famous Yamabe problem. While the situation is well understood for compact manifolds, things are less clear when dealing with complete manifolds. Aubin proved in '76 that optimal Sobolev inequalities are valid for complete manifolds with bounded sectional curvature and positive injectivity radius. We prove here that the result still holds if the bound on the sectional curvature is replaced by a lower bound on the Ricci curvature (a much weaker assumption). We also get estimates for the remaining constants.

Riemann compatible tensors

Abstract: Derdzinski and Shen's theorem on the restrictions posed by a Codazzi tensor on the Riemann tensor holds more generally when a Riemann-compatible tensor exists. Several properties are shown to remain valid in this broader setting. Riemann compatibility is equivalent to the Bianchi identity of the new "Codazzi deviation tensor" with a geometric significance. Examples are given of manifolds with Riemann-compatible tensors, in particular those generated by geodesic mapping. Compatibility is extended to generalized curvature tensors with an application to Weyl's tensor and general relativity.

Variational Approach to Differential Invariants of Rank 2 Vector Distributions

Abstract: In the present paper we construct differential invariants for generic rank 2 vector distributions on n-dimensional manifold. In the case n = 5 (the first case containing functional parameters) E. Cartan found in 1910 the covariant fourth-order tensor invariant for such distributions, using his ”reduction-prolongation” procedure (see [12]). After Cartan’s work the following questions remained open: first the geometric reason for existence of Cartan’s tensor was not clear; secondly it was not clear how to generalize this tensor to other classes of distributions; finally there were no explicit formulas for computation of Cartan’s tensor. Our paper is the first in the series of papers, where we develop an alternative approach, which gives the answers to the questions mentioned above. It is based on the investigation of dynamics of the field of so-called abnormal extremals (singular curves) of rank 2 distribution and on the general theory of unparametrized curves in the Lagrange Grassmannian, developed in [4],[5]. In this way we construct the fundamental form and the projective Ricci curvature of rank 2 vector distributions for arbitrary n ≥ 5. For n = 5 we give an explicit method for computation of these invariants and demonstrate it on several examples. In the next paper [19] we show that in the case n = 5 our fundamental form coincides with Cartan’s tensor.