Topic
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.
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01 Aug 1991
TL;DR: This paper describes a formulation which gives bounded non-zero curvature and close to curvature continuity and although it was claimed at the conference that this formulation was C2, this is not (quite) true.
Abstract: Previous cubic recursive division schemes have had either zero or unbounded curvature at the singular points. This paper describes a formulation which gives bounded non-zero curvature and close to curvature continuity. Although it was claimed at the conference that this formulation was C2, this is not (quite) true.
55 citations
TL;DR: In this paper, the authors formulate a nonlinear theory of thermal stresses and obtain the explicit form of the governing partial differential equations for this equilibrium change, and show that geometric linearization leads to governing equations that are identical to those of the classical linear theory.
Abstract: In this paper we formulate a geometric theory of thermal stresses. Given a temperature distribution, we associate a Riemannian material manifold to the body, with a metric that explicitly depends on the temperature distribution. A change of temperature corresponds to a change of the material metric. In this sense, a temperature change is a concrete example of the so-called referential evolutions. We also make a concrete connection between our geometric point of view and the multiplicative decomposition of deformation gradient into thermal and elastic parts. We study the stress-free temperature distributions of the finite-deformation theory using curvature tensor of the material manifold. We find the zero-stress temperature distributions in nonlinear elasticity. Given an equilibrium configuration, we show that a change of the material manifold, i.e. a change of the material metric will change the equilibrium configuration. In the case of a temperature change, this means that given an equilibrium configuration for a given temperature distribution, a change of temperature will change the equilibrium configuration. We obtain the explicit form of the governing partial differential equations for this equilibrium change. We also show that geometric linearization of the present nonlinear theory leads to governing equations that are identical to those of the classical linear theory of thermal stresses.
55 citations
TL;DR: In this article, the general solution for a symmetric second-order tensor of the Riemann tensor is given in terms of the curvature 2-form structure of a space-time manifold.
Abstract: The general solution for a symmetric second-order tensorX of the equationX
e(a
R
e
b
cd=0 whereR is the Riemann tensor of a space-time manifold, andX is obtained in terms of the curvature 2-form structure ofR by a straightforward geometrical technique, and agrees with that given by McIntosh and Halford using a different procedure. Two results of earlier authors are derived as simple corollaries of the general theorem.
54 citations
Book•
01 Dec 2011
TL;DR: In this article, the Riemann curvature tensor tensor is used to measure the curvature of smooth manifolds, and the Hodge star is used as a measure of curvature.
Abstract: 1 Smooth manifolds 2 Matrices and Lie groups 3 Introduction to vector bundles 4 Algebra of vector bundles 5 Maps and vector bundles 6 Vector bundles with fiber C]n 7 Metrics on vector bundles 8 Geodesics 9 Properties of geodesics 10 Principal bundles 11 Covariant derivatives and connections 12 Covariant derivatives, connections and curvature 13 Flat connections and holonomy 14 Curvature polynomials and characteristic classes 15 Covariant derivatives and metrics 16 The Riemann curvature tensor 17 Complex manifolds 18 Holomorphic submanifolds, holomorphic sections and curvature 19 The Hodge star Indexed list of propositions by subject Index
54 citations
TL;DR: This research converts convex integration theory into an algorithm that produces isometric maps of flat tori and demonstrates that convex Integration, a theory still confined to specialists, can produce computationally tractable solutions of partial differential relations.
Abstract: It is well-known that the curvature tensor is an isometric invariant of C2 Riemannian manifolds. This invariant is at the origin of the rigidity observed in Riemannian geometry. In the mid 1950s, Nash amazed the world mathematical community by showing that this rigidity breaks down in regularity C1. This unexpected flexibility has many paradoxical consequences, one of them is the existence of C1 isometric embeddings of flat tori into Euclidean three-dimensional space. In the 1970s and 1980s, M. Gromov, revisiting Nash’s results introduced convex integration theory offering a general framework to solve this type of geometric problems. In this research, we convert convex integration theory into an algorithm that produces isometric maps of flat tori. We provide an implementation of a convex integration process leading to images of an embedding of a flat torus. The resulting surface reveals a C1 fractal structure: Although the tangent plane is defined everywhere, the normal vector exhibits a fractal behavior. Isometric embeddings of flat tori may thus appear as a geometric occurrence of a structure that is simultaneously C1 and fractal. Beyond these results, our implementation demonstrates that convex integration, a theory still confined to specialists, can produce computationally tractable solutions of partial differential relations.
54 citations