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.

Fermi Normal Coordinates and Some Basic Concepts in Differential Geometry

Abstract: Fermi coordinates, where the metric is rectangular and has vanishing first derivatives at each point of a curve, are constructed in a particular way about a geodesic. This determines an expansion of the metric in powers of proper distance normal to the geodesic, of which the second‐order terms are explicitly computed here in terms of the curvature tensor at the corresponding point on the base geodesic. These terms determine the lowest‐order effects of a gravitational field which can be measured locally by a freely falling observer. An example is provided in the Schwarzschild metric. This discussion of Fermi Normal Coordinate provides numerous examples of the use of the modern, coordinate‐free concept of a vector and of computations which are simplified by introducing a vector instead of its components. The ideas of contravariant vector and Lie Bracket, as well as the equation of geodesic deviation, are reviewed before being applied.

Manifolds with 1/4-pinched curvature are space forms

Abstract: Let (M,g_0) be a compact Riemannian manifold with pointwise 1/4-pinched sectional curvatures. We show that the Ricci flow deforms g_0 to a constant curvature metric. The proof uses the fact, also established in this paper, that positive isotropic curvature is preserved by the Ricci flow in all dimensions. We also rely on earlier work of Hamilton and of Bohm and Wilking.

Electrodynamics in the general relativity theory

Abstract: The restricted relativity theory resulted mathematically in the introduction of pseudo-euclidean four-dimensional space and the welding together of the electric and magnetic force vectors into the electromagnetic tensor. Einstein's general relativity theory led to the assumption that the fourdimensional space mentioned above is a curved space and the curvTature was made to account for the gravitational phenomena. The Riemann tensor which measures the curvature and the electromagnetic tensor seem thus to play essentially different roles in physics: the former reflects some properties of the space so that gravitation may be said to have been geometricized,-when the space is given all the gravitational features are determined; on the contrary, it seemed that the electromagnetic tensor is superposed on the space, that it is something external with respect to the space, that after space is given the electromagnetic tensor can be given in different ways. Several attempts were made to geometricize the electromagnetic forces, to find a geometric interpretation for the electromagnetic tensor, to inicorporate this tensor into the space in the sense in which the gravitational forces had been incorporated. It seemed that in order to do this it was necessary to change the geometry; to abandon the Riemanii geometry and to adopt a more general space with a more complicated curvature tensor, one part of which would then account for the gravitational properties and the other wouild in the same way account for the electromagnetic phenomena. H. Weyl arrived in a most natural way to such a generalization. His theory always will remaini a brilliant mathematical feat, but it seems that it did not fulfil the expectationis as a physical theory and the same seems to be true with respect to other attempts. The electromagnetic tensor is, however, niot elntirely independelnt of the Riemann tensor in the ordinary genieral relativity theory; these two tenisors are connected by the so called eniergy relation; it seemed to be desirable to try, witliout breaking the frame of the Riemann geometry, to study

Lectures on the Ricci Flow

Abstract: 1. Introduction 2. Riemannian geometry background 3. The maximum principle 4. Comments on existence theory for parabolic PDE 5. Existence theory for the Ricci flow 6. Ricci flow as a gradient flow 7. Compactness of Riemannian manifolds and flows 8. Perelman's W entropy functional 9. Curvature pinching and preserved curvature properties under Ricci flow 10. Three-manifolds with positive Ricci curvature and beyond.