Related papers: On equivalency of various geometric structures

Inspired by recent work of Kopparty-Moshkovitz-Zuiddam and motivated by problems in combinatorics and hypergraphs, we introduce the notion of the symmetric geometric rank of a symmetric tensor. This quantity is equal to the codimension of…

A regularization procedure, that allows one to relate singularities of curvature to those of the Einstein tensor without some of the shortcomings of previous approaches, is proposed. This regularization is obtained by requiring that (i) the…

From the Aharonov-Bohm effect to general relativity, geometry plays a central role in modern physics. In quantum mechanics many physical processes depend on the Berry curvature. However, recent advances in quantum information theory have…

We describe and construct here pseudo-Hermitian structures $\theta$ without torsion (i.e. with transversal symmetry) whose Webster-Ricci curvature tensor is a constant multiple of the exterior differential $d\theta$. We call these…

The local geometry of a Riemannian symmetric space is described completely by the Riemannian metric and the Riemannian curvature tensor of the space. In the present article I describe how to compute these tensors for any Riemannian…

A spacetime is a connected 4-dimensional semi-Riemannian manifold endowed with a metric $g$ with signature $(- + + +)$. The geometry of a spacetime is described by the metric tensor $g$ and the Ricci tensor $S$ of type $(0, 2)$ whereas the…

In this paper, we study several types of geometric problems related to the Ricci curvature on noncompact complex manifolds, such as the existence of K\"{a}hler-Einstein metrics on complete K\"{a}hler manifolds with negative Ricci curvature,…

An expansion is developed for the Weil-Petersson Riemann curvature tensor in the thin region of the Teichm\"{u}ller and moduli spaces. The tensor is evaluated on the gradients of geodesic-lengths for disjoint geodesics. A precise lower…

We propose a new graph kernel for graph classification and comparison using Ollivier Ricci curvature. The Ricci curvature of an edge in a graph describes the connectivity in the local neighborhood. An edge in a densely connected…

In this work a proposal for definition of twistors on generic curved spaces is exposed and investigated. We consider superpositions of nearly autoparallel and nearly geodesic maps (nearly conformal maps, nc-maps) of (pseudo-)Riemannian…

The paper introduces the class of O-metric spaces, a novel generalization of metric-type spaces, classifying almost all possible metric types into upward and downward O-metrics. We list some topologies arising from O-metrics and discuss…

According to the teleparallel equivalent of general relativity, curvature and torsion are two equivalent ways of describing the same gravitational field. Despite equivalent, however, they act differently: whereas curvature yields a…

We consider hierarchical structures such as Fibonacci sequences and Penrose tilings, and examine the consequences of different choices for the definition of isomorphism. In particular we discuss the role such a choice plays with regard to…

We make a geometric study of the Geometric Rank of tensors recently introduced by Kopparty et al. Results include classification of tensors with degenerate geometric rank in $C^3\otimes C^3\otimes C^3$, classification of tensors with…

An algebraic curvature tensor is called Osserman if the eigenvalues of the associated Jacobi operator are constant on the unit sphere. A Riemannian manifold is called conformally Osserman if its Weyl conformal curvature tensor at every…

In this paper, the Cartan tensors of the $(\alpha,\beta)$-norms are investigated in details. Then an equivalence theorem of $(\alpha,\beta)$-norms is proved. As a consequence in Finsler geometry, general $(\alpha,\beta)$-metrics on smooth…

In this paper we present several curvature estimates and convergence results for solutions of the Ricci flow. The curvature estimates depend on smallness of certain local space-time integrals of the norm of the Riemann curvature tensor,…

In the class of metrics of a generic conformal structure there exists a distinguishing metric. This was noticed by Albert Einstein in a lesser-known paper of 1921 (Berl. Ber., 1921, pp. 261-264). We explore this finding from a geometrical…

Connections between continuous and discrete worlds tend to be elusive. One example is curvature. Even though there exist numerous nonequivalent definitions of graph curvature, none is known to converge in any limit to any traditional…

For a noncollapsed Gromov-Hausdorff convergent sequence of Riemannian manifolds with a uniform bound of Ricci curvature, we establish two spectral convergence. One of them is on the Hodge Laplacian acting on differential one-forms. The…