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.

Manifolds of Positive Scalar Curvature: A Progress Report

Abstract: In the special case n = 2, the scalar curvature is just twice the Gaussian curvature. This paper will deal with bounds on the scalar curvature, and especially, with the question of when a given manifold (always assumed C∞) admits a Riemannian metric with positive or non-negative scalar curvature. (If the manifold is non-compact, we require the metric to be complete; otherwise this is no restriction at all.) We will not go over the historical development of this subject or everything that is known about it; instead, our focus here will be on updating the existing surveys [20], [68], [69] and [58].

Multiplicative functional for the heat equation on manifolds with boundary

Abstract: By the Weitzenbock formula relating the Hodge–de Rham Laplacian and the covariant Laplacian for differential forms on a Riemannian manifold, the heat equation for differential forms is naturally associated with a matrix-valued Feynman– Kac multiplicative functional determined by the curvature tensor. The case of a closed manifold (without boundary) is well known and will be briefly reviewed below. In constrast, the case of manifolds with boundary is not well known, and for good reasons. Because the absolute boundary condition on differential forms is Dirichlet in the normal direction and Neumann in the tangential directions, the associated multiplicative functional is discontinuous and much more difficult to handle. Ikeda and Watanabe [6; 7] have dealt with this situation by using an excursion theory (for reflecting Brownian motion) that seems to have been created especially for this problem. In this paper we suggest a different approach that is based on an idea of approximation due to Airault [1]. This construction has the advantage that a key property of the multiplicative functional (i.e., the attendant Ito’s formula for this functional) follows almost automatically from the approximate multiplicative functional without resorting to excursion theory, thus greatly simplifying this part of the theory; see Theorem 3.7. Before coming to another and more important raison d’etre for the present work, we briefly review some relevant facts for a closed manifold. LetM be a compact, closed Riemannian manifold and let α0 be a 1-form onM. Consider the following initial value problem:  ∂α ∂t = 1 2 α,

Detection of the gravitomagnetic field using an orbiting superconducting gravity gradiometer. Theoretical principles.

Abstract: The angular momentum of the Earth produces gravitomagnetic components of the Riemann curvature tensor, which are of the order of ${10}^{\mathrm{\ensuremath{-}}10}$ of the Newtonian tidal terms arising from the mass of the Earth. These components could be detected in principle by sensitive superconducting gravity gradiometers currently under development. We lay out the theoretical principles of such an experiment by using the parametrized post-Newtonian formalism to derive the locally measured Riemann tensor in an orbiting proper reference frame, in a class of metric theories of gravity that includes general relativity. A gradiometer assembly consisting of three gradiometers with axes at mutually right angles measures three diagonal components of a 3\ifmmode\times\else\texttimes\fi{}3 ``tidal tensor,'' related to the Riemann tensor. We find that, by choosing a particular assembly orientation relative to the orbit and taking a sum and difference of two of the three gradiometer outputs, one can isolate the gravitomagnetic relativistic effect from the large Newtonian background.