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.

Osserman manifolds of dimension 8

Abstract: For a Riemannian manifold Mn with the curvature tensor R, the Jacobi operator RX is defined by RXY=R(X,Y)X. The manifold Mn is called pointwise Osserman if, for every p ∈ Mn, the eigenvalues of the Jacobi operator RX do not depend of a unit vector X ∈ TpMn, and is called globally Osserman if they do not depend of the point p either. R. Osserman conjectured that globally Osserman manifolds are flat or locally rank-one symmetric. This Conjecture is true for manifolds of dimension n≠8,16[14]. Here we prove the Osserman Conjecture and its pointwise version for 8-dimensional manifolds.

Conformal Geometry from the Riemannian Viewpoint

Abstract: When a conformai structure on a manifold is defined by a Riemannian metric g , how to detect conformai flatness on g ? The answer, due to Weyl and Schouten, is given in § C, and some applications are derived in § D. It turns out that the three dimensional case, i.e. the case where the curvature tensor is determined by the Ricci tensor, needs a special treatment. An example of that situation is given in § E. We also give some global properties of compact conformally flat manifolds: the nullity of their Pontryagin numbers, (Chern-Simons), a vanishing theorem for middle-dimensional cohomology when the scalar curvature is positive (Bourguignon)and a structure’ theorem when the scalar curvature is zero.

Remarks on Kähler-Einstein manifolds

Abstract: The main purpose of this note is to characterize a compact Kahler-Einstein manifold in terms of curvature form The curvature form Q is an EndT valued differential form of type (1,1) which represents the curvature class of the manifold We shall prove that the curvature form of a Kahler metric is the harmonic representative of the curvature class if and only if the Kahler metric is an Einstein metric in the generalized sense (gs), that is, if the Ricci form of the metric is parallel It is well known that a Kahler metric is an Einstein metric in the g s if and only if it is locally product (globally, if the manifold is simply connected and complete) of Kahler-Einstein metrics We obtain an integral formula, involving the integral of the trace of some operators defined by the curvature tensor, which measures the deviation of a Kahler-Einstein metric from a Hermitian symmetric metric In the final section we shall prove the uniqueness up to equivalence of Kahler-Einstein metrics in a simply connected compact complex homogeneous space This result was proved by Berger in the case of a complex projective space and our proof is completely different from Berger’s

Robust estimation of adaptive tensors of curvature by tensor voting

Abstract: Although curvature estimation from a given mesh or regularly sampled point set is a well-studied problem, it is still challenging when the input consists of a cloud of unstructured points corrupted by misalignment error and outlier noise. Such input is ubiquitous in computer vision. In this paper, we propose a three-pass tensor voting algorithm to robustly estimate curvature tensors, from which accurate principal curvatures and directions can be calculated. Our quantitative estimation is an improvement over the previous two-pass algorithm, where only qualitative curvature estimation (sign of Gaussian curvature) is performed. To overcome misalignment errors, our improved method automatically corrects input point locations at subvoxel precision, which also rejects outliers that are uncorrectable. To adapt to different scales locally, we define the RadiusHit of a curvature tensor to quantify estimation accuracy and applicability. Our curvature estimation algorithm has been proven with detailed quantitative experiments, performing better in a variety of standard error metrics (percentage error in curvature magnitudes, absolute angle difference in curvature direction) in the presence of a large amount of misalignment noise.

Noether symmetries in Gauss–Bonnet-teleparallel cosmology

Abstract: A generalized teleparallel cosmological model, [Formula: see text], containing the torsion scalar T and the teleparallel counterpart of the Gauss-Bonnet topological invariant [Formula: see text], is studied in the framework of the Noether symmetry approach. As [Formula: see text] gravity, where [Formula: see text] is the Gauss-Bonnet topological invariant and R is the Ricci curvature scalar, exhausts all the curvature information that one can construct from the Riemann tensor, in the same way, [Formula: see text] contains all the possible information directly related to the torsion tensor. In this paper, we discuss how the Noether symmetry approach allows one to fix the form of the function [Formula: see text] and to derive exact cosmological solutions.