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.

Convergence of riemannian manifolds with integral bounds on curvature. I

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Interface Evolution in Three Dimensions¶with Curvature-Dependent Energy¶and Surface Diffusion:¶Interface-Controlled Evolution, Phase Transitions, Epitaxial Growth of Elastic Films

Abstract: When the interfacial energy is a nonconvex function of orientation, the anisotropic-curvature-flow equation becomes backward parabolic. To overcome the instability thus generated, a regularization of the equation that governs the evolution of the interface is needed. In this paper we develop a regularized theory of curvature flow in three dimensions that incorporates surface diffusion and bulk-surface interactions. The theory is based on a superficial mass balance; configurational forces and couples consistent with superficial force and moment balances; a mechanical version of the second law that includes, via the configurational moments, work that accompanies changes in the curvature of the interface; a constitutive theory whose main ingredient is a positive-definite, isotropic, quadratic dependence of the interfacial energy on the curvature tensor. Two special cases are investigated: (i) the interface is a boundary between bulk phases or grains, and (ii) the interface separates an elastic thin film bonded to a rigid substrate from a vapor phase whose sole action is the deposition of atoms on the surface.

Type D Einstein spacetimes in higher dimensions

Abstract: We show that all static spacetimes in higher dimensions n > 4 are necessarily of Weyl types G, Ii, D or O. This also applies to stationary spacetimes provided additional conditions are fulfilled, as for most known black hole/ring solutions. (The conclusions change when the Killing generator becomes null, such as at Killing horizons, on which we briefly comment.) Next we demonstrate that the same Weyl types characterize warped product spacetimes with a one-dimensional Lorentzian (timelike) factor, whereas warped spacetimes with a two-dimensional Lorentzian factor are restricted to the types D or O. By exploring algebraic consequences of the Bianchi identities, we then analyze the simplest non-trivial case from the above classes?type D vacuum spacetimes, possibly with a cosmological constant, dropping, however, the assumptions that the spacetime is static, stationary or warped. It is shown that for 'generic' type D vacuum spacetimes (as defined in the text) the corresponding principal null directions are geodetic in arbitrary dimension (this in fact also applies to type II spacetimes). For n ? 5, however, there may exist particular cases of type D vacuum spacetimes which admit non-geodetic multiple principal null directions and we explicitly present such examples in any n ? 7. Further studies are restricted to five dimensions, where the type D Weyl tensor is fully described by a 3 ? 3 real matrix ?ij. In the case with 'twistfree' (Aij = 0) principal null geodesics we show that in a 'generic' case ?ij is symmetric and eigenvectors of ?ij coincide with eigenvectors of the expansion matrix Sij providing us thus in general with three preferred spacelike directions of the spacetime. Similar results are also obtained when relaxing the twistfree condition and assuming instead that ?ij is symmetric. The five-dimensional Myers?Perry black hole and Kerr?NUT?AdS metrics in arbitrary dimension are also briefly studied as specific illustrative examples of type D vacuum spacetimes.

Flows of G_2-structures, I

Abstract: This is a foundational paper on flows of G_2 Structures. We use local coordinates to describe the four torsion forms of a G_2 Structure and derive the evolution equations for a general flow of a G_2 Structure on a 7-manifold. Specifically, we compute the evolution of the metric, the dual 4-form, and the four independent torsion forms. In the process we obtain a simple new proof of a theorem of Fernandez-Gray. As an application of our evolution equations, we derive an analogue of the second Bianchi identity in G_2-geometry which appears to be new, at least in this form. We use this result to derive explicit formulas for the Ricci tensor and part of the Riemann curvature tensor in terms of the torsion. These in turn lead to new proofs of several known results in G_2 geometry.

Beyond Einstein-Cartan gravity: quadratic torsion and curvature invariants with even and odd parity including all boundary terms

Abstract: Recently, gravitational gauge theories with torsion have been discussed by an increasing number of authors from a classical as well as from a quantum field theoretical point of view. The Einstein–Cartan(–Sciama–Kibble) Lagrangian has been enriched by the parity odd pseudoscalar curvature (Hojman, Mukku and Sayed) and by torsion square and curvature square pieces, likewise of even and odd parity. (i) We show that the inverse of the so-called Barbero–Immirzi parameter multiplying the pseudoscalar curvature, because of the topological Nieh–Yan form, can be appropriately discussed if torsion square pieces are included. (ii) The quadratic gauge Lagrangian with both parities, proposed by Obukhov et al and Baekler et al, emerges also in the framework of Diakonov et al. We establish the exact relations between both approaches by applying the topological Euler and Pontryagin forms in a Riemann–Cartan space expressed for the first time in terms of irreducible pieces of the curvature tensor. (iii) In a Riemann–Cartan spacetime, that is, in a spacetime with torsion, parity-violating terms can be brought into the gravitational Lagrangian in a straightforward and natural way. Accordingly, Riemann–Cartan spacetime is a natural habitat for chiral fermionic matter fields.