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.

Second fundamental measure of geometric sets and local approximation of curvatures

Abstract: Using the theory of normal cycles, we associate with each geometric subset of a Riemannian manifold a —tensor-valued— curvature measure, which we call its second fundamental measure. This measure provides a finer description of the geometry of singular sets than the standard curvature measures. Moreover, we deal with approximation of curvature measures. We get a local quantitative estimate of the difference between curvature measures of two geometric subsets, when one of them is a smooth hypersurface.

Compatibility Equations of Nonlinear Elasticity for Non-Simply-Connected Bodies

Abstract: Compatibility equations of elasticity are almost 150 years old. Interestingly, they do not seem to have been rigorously studied, to date, for non-simply-connected bodies. In this paper we derive necessary and sufficient compatibility equations of nonlinear elasticity for arbitrary non-simply-connected bodies when the ambient space is Euclidean. For a non-simply-connected body, a measure of strain may not be compatible, even if the standard compatibility equations (“bulk” compatibility equations) are satisfied. It turns out that there may be topological obstructions to compatibility; this paper aims to understand them for both deformation gradient F and the right Cauchy-Green strain C = F T F. We show that the necessary and sufficient conditions for compatibility of deformation gradient F are the vanishing of its exterior derivative and all its periods, that is, its integral over generators of the first homology group of the material manifold. We will show that not every non-null-homotopic path requires supplementary compatibility equations for F and linearized strain e. We then find both necessary and sufficient compatibility conditions for the right Cauchy-Green strain tensor C for arbitrary non-simply-connected bodies when the material and ambient space manifolds have the same dimensions. We discuss the well-known necessary compatibility equations in the linearized setting and the Cesaro-Volterra path integral. We then obtain the sufficient conditions of compatibility for the linearized strain when the body is not simply-connected. To summarize, the question of compatibility reduces to two issues: i) an integrability condition, which is d(F dX) = 0 for the deformation gradient and a curvature vanishing condition for C, and ii) a topological condition. For F dx this is a homological condition because the equation one is trying to solve takes the form dφ = F dX. For C, however, parallel transport is involved, which means that one needs to solve an equation of the form dR/ ds = RK, where R takes values in the orthogonal group. This is, therefore, a question about an orthogonal representation of the fundamental group, which, as the orthogonal group is not commutative, cannot, in general, be reduced to a homological question.

Gravity from non-commutative geometry

Abstract: We introduce the linear connection in the non-commutative geometry model of the product of continuous manifold and the discrete space of two points. We discuss its metric properties, define the metric connection and calculate the curvature. We define also the Ricci tensor and the scalar curvature. We find that the latter differs from the standard scalar curvature of the manifold by a term, which might be interpreted as the cosmological constant, and apart from that we find no other dynamic fields in the model. Finally we discuss an example solution of flat linear connection, with the non-trivial scaling dependence of the metric tensor on the discrete variable. We interpret the obtained solution as confirmed by the standard model, with the scaling factor corresponding to the Weinberg angle.

Curvature properties of twistor spaces of quaternionic Kähler manifolds

Abstract: We present a study of natural almost Hermitian structures on twistor spaces of quaternionic Kahler manifolds. This is used to supply (4n + 2)-dimensional examples (n > 1) of symplec tic non-Kahler manifolds. Studying their curvature properties we give a negative answer to the questions raised by D.Blair-S.Ianus and A.Gray, respectively, of whether a compact almost Kahler manifold with Hermitian Ricci tensor or whose curvature tensor belongs to the class AH2 is Kahler.