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.

Curvature and global rigidity in finsler manifolds

Abstract: We present some strong global rigidity results for reversible Finsler manifolds. Following ‡ Cartan's de‹nition (1926), a locally symmet- ric Finsler metric is one whose curvature is parallel. These spaces strictly contain the spaces such that the geodesic re›ections are local isometries and also constant curvature manifolds. In the case of negative curvature, we prove that the locally symmetric Finsler metrics on compact manifolds are Riemannian and this, therefore, extends A. Zadeh's rigidity result. Our ap- proach uses dynamical properties of the ›ag curvature. We also give a full generalization of the Ossermann Sarnak minoration of the metric entropy of the geodesic ›ow. In positive curvature, we just announce some partial results and remarks concerning Finsler metrics of curvature +1 on the 2- sphere. We show that in the reversible case the geodesic ›ow is conjugate to the standard one. We also observe that a condition of integral geometry (of Radon type) forces such a metric to be Riemannian. This indicates a deep link with (exotic) projective structures.

Eikonal scattering and asymptotic superluminality of massless higher spin fields

Abstract: We consider scattering of massless higher-spin particles in the eikonal regime in four dimensions. By demanding the absence of asymptotic superluminality, corresponding to positivity of the eikonal phase, we place constraints on the possible cubic couplings which can appear in the theory. The cubic couplings come in two types: lower-derivative non-Abelian vertices and higher-derivative Abelian vertices made out of gauge-invariant curvature tensors. We find that the Abelian couplings between massless higher spins lead to an asymptotic time advance for certain choices of polarizations, indicating that these couplings should be absent unless new states come in at the scale suppressing the derivatives in these couplings. A subset of non-Abelian cubic couplings are consistent with eikonal positivity, but are ruled out by consistency of the four-particle amplitude away from the eikonal limit. The eikonal constraints are, therefore, complementary to the four-particle test, ruling out even trivial cubic curvature couplings in any theory with a finite number of massless higher spins and no new physics at the scale suppressing derivatives in these vertices.

Adjugate Diffusion Tensors for Geodesic Tractography in White Matter

Abstract: One of the approaches in diffusion tensor imaging is to consider a Riemannian metric given by the inverse diffusion tensor. Such a metric is used for geodesic tractography and connectivity analysis in white matter. We propose a metric tensor given by the adjugate rather than the previously proposed inverse diffusion tensor. The adjugate metric can also be employed in the sharpening framework. Tractography experiments on synthetic and real brain diffusion data show improvement for high-curvature tracts and in the vicinity of isotropic diffusion regions relative to most results for inverse (sharpened) diffusion tensors, and especially on real data. In addition, adjugate tensors are shown to be more robust to noise.

On projective K\"ahler manifolds of partially positive curvature and rational connectedness

Abstract: In a previous paper, we proved that a projective Kahler manifold of positive total scalar curvature is uniruled. At the other end of the spectrum, it is a well-known theorem of Campana and Kollar-Miyaoka-Mori that a projective Kahler manifold of positive Ricci curvature is rationally connected. In the present work, we investigate the intermediate notion of $k$-positive Ricci curvature and prove that for a projective $n$-dimensional Kahler manifold of $k$-positive Ricci curvature the MRC fibration has generic fibers of dimension at least $n-k+1$. We also establish an analogous result for projective Kahler manifolds of semi-positive holomorphic sectional curvature based on an invariant which records the largest codimension of maximal subspaces in the tangent spaces on which the holomorphic sectional curvature vanishes. In particular, the latter result confirms a conjecture of S.-T. Yau in the projective case.