Locally homogeneous non-gradient quasi-Einstein 3-manifolds

In this paper, we classify the compact locally homogeneous non-gradient m-quasi Einstein 3- manifolds. Along the way, we also prove that given a compact quotient of a Lie group of any dimension that is m-quasi Einstein, the potential vector field X must be left invariant and Killing. We also classify the nontrivial m-quasi Einstein metrics that are a compact quotient of the product of two Einstein metrics. We also show that S1 is the only compact manifold of any dimension which admits a metric which is nontrivially m-quasi Einstein and Einstein.

Ricci almost solitons with associated projective vector field

A Ricci almost soliton whose associated vector field is projective is shown to have vanishing Cotton tensor, divergence-free Bach tensor and Ricci tensor as conformal Killing. For the compact case, a sharp inequality is obtained in terms of scalar curvature.We show that every complete gradient Ricci soliton is isometric to the Riemannian product of a Euclidean space and an Einstein space. A complete K-contact Ricci almost soliton whose associated vector field is projective is compact Einstein and Sasakian.

The flow of the Berry curvature vector field

The concept of Berry phase and Berry curvature has become ubiquitous in solid state physics as it relates to variety of phenomena, such as topological insulators, polarization, and various Hall effects. It is well known that large Berry curvatures arise from close proximity of hybridizing bands, however, the vectorial nature of the Berry curvature is not utilized in current research. On bulk bcc Fe, we demonstrate the flow of the Berry curvature vector field which features not only monopoles but also higher dimensional structures with its own topological features. They can provide a novel unique view on the electronic structure in all three dimensions. This knowledge is also used to quantify particular contributions to the intrinsic anomalous Hall effect in a simple analytical form.

The Oriented and Flux-Weighted Current Density Stagnation Graph of LiH

A scheme is introduced to quantitatively analyze the magnetically induced molecular current density vector field $\mathbf{J}$. After determining the set of zero points of $\mathbf{J}$, which is called its {\em stagnation graph} (SG), the line integrals $\Phi_{\ell_i}=-\frac{1}{\mu_0} \int_{\ell_i} \mathbf{B}_\mathrm{ind}\cdot\mathrm{d}\mathbf{l}$ along all edges $\ell_i$ of the connected subset of the SG are determined. The edges $\ell_i$ are oriented such that all $\Phi_{\ell_i}$ are non-negative and they are weighted with $\Phi_{\ell_i}$. An oriented flux-weighted (current density) stagnation graph (OFW-SG) is obtained. Since $\mathbf{J}$ is in the exact theoretical limit divergence free and due to the topological characteristics of such vector fields the flux of all separate vortices and neighbouring vortex combinations can be determined by adding the weights of cyclic subsets of edges of the OFW-SG. The procedure is exemplified by the case of LiH for a perpendicular and weak homogeneous external magnetic field $\mathbf{B}$}

Certain results on trans-paraSasakian 3-manifolds

Let M be a trans-paraSasakian 3-manifold. In this paper, the necessary and sufficient condition for the Reeb vector field of a trans-paraSasakian 3-manifold to be harmonic is obtained. Also, it is proved that the Ricci operator of M is invariant along the Reeb flow if and only if M is a paracosymplectic manifold, an α-paraSasakian manifold or a space of negative constant sectional curvature.