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.

Lorentzian $4$ dimensional manifolds with ``local isotropy''

Abstract: We define “locally isotropic” spaces, as spaces in which there exists, in the tangent space at each pointP, a subgroupA (P) (of dimension at least 1) of the Lorentz groupL + ↑ , leaving the Riemann tensor and its 2 first covariant derivatives invariant; the subgroupsA(P) are assumed to be conjugate inL + ↑ . These spaces admit a group of local isometriesG. IfI P denotes the subgroup ofG leavingP fixed, thendA (P)=I P . All spaces of petrov type D, admitting local isotropy are determined.

A new modified couple stress theory for anisotropic elasticity and microscale laminated Kirchhoff plate model

Abstract: A new modified couple stress theory for anisotropic elasticity is proposed. This theory contains three material length scale parameters. Differing from the modified couple stress theory, the couple stress constitutive relationships are introduced for anisotropic elasticity, in which the curvature (rotation gradient) tensor is asymmetric and the couple stress moment tensor is symmetric. However, under isotropic case, this theory can be identical to modified couple stress theory proposed by Yang et al. (Int J Solids Struct 39:2731–2743, 2002). The differences and relations of standard, modified and new modified couple stress theories are given herein. A detailed variational formulation is provided for this theory by using the principle of minimum total potential energy. Based on the new modified couple stress theory, composite laminated Kirchhoff plate models are developed in which new anisotropic constitutive relationships are defined. The First model contains two material length scale parameters, one related to fiber and the other related to matrix. The curvature tensor in this model is asymmetric; however, the couple stress moment tensor is symmetric. Under isotropic case, this theory can be identical to the modified couple stress theory proposed by Yang et al. (Int J Solids Struct 39:2731–2743, 2002). The present model can be viewed as a simplified couple stress theory in engineering mechanics. Moreover, a more simplified model of couple stress theory including only one material length scale parameter for modeling the cross-ply laminated Kirchhoff plate is suggested. Numerical results show that the proposed laminated Kirchhoff plate model can capture the scale effects of microstructures.

Construction of manifolds of positive scalar curvature

Abstract: We prove that a regular neighborhood of a codimension > 3 subcomplex of a manifold can be chosen so that the induced metric on its boundary has positive scalar curvature. A number of useful facts concerning manifolds of positive scalar curvature follow from this construction. For example, we see that any finitely presented group can appear as the fundamental group of a compact 4-manifold with such a metric. 0. Outline of results. We give a new method for constructing complete Riemannian manifolds of positive scalar curvature and use it to continue the investigation of properties of positive scalar curvature. Our construction uses the idea that manifolds having spheres of dimension > 2 as "factors" will admit metrics of positive scalar curvature if the spheres can be made to carry sufficient positive curvature to dominate any negative curvature. Most of the known methods for constructing manifolds of positive scalar curvature employ this same idea. For example, any manifold of the form M X S2 can be given a warped-product metric of positive scalar curvature by suitably adjusting the radius of the S2-factor. Similarly, by deforming the standard metric on S3-{point} in a small neighborhood of the point and using the S2-factor to carry positive curvature around the corner we can construct a complete metric of positive scalar curvature on R3. This same idea was used by Gromov and Lawson [GL] and Schoen and Yau [SY] in proving that codimension > 3 surgeries on a manifold of positive scalar curvature yields a manifold which also carries positive scalar curvature. In this paper we generalize the above techniques to cover any manifold formed as the boundary of a regular neighborhood of a subcomplex K of a manifold M. If the codimension of K > 3 this boundary looks locally like K X S2, and so should carry positive scalar curvature. THEOREM 1. Let M be an n-dimensional Riemannian manifold with a fixed smooth cell decomposition and K a codimension q > 3 subcomplex of M. Then there is a regular neighborhood U of K in M so that the induced metric on the boundary dU has positive scalar curvature. An easy consequence of this theorem is the following. COROLLARY 2. Let 7r be a finitely presented group. Then there exists a compact 4-manifold M of positive scalar curvature with 7rl (M) = 7r. This fact is interesting since it is generally believed that manifolds that are "large" in some sense should not adrrlit metrics of positive curvature. Corollary 2 Receiv?d by the editors Septem})er 25, 1985 and, ill revised form, July 17, 1986. 1980 M(lthf'rB(lti('.s.S?l{Jjf'('t (l(l.N'.N'iJl('(ltiOn (1985 RfviSion). Primary 53C20. (r)1988 Americatl Mathematic.al Society 0002-9947/88 $1.00 + $.25 per page