arXiv research

Proposes a new nonlocal curvature tensor concept.

In this survey, we study three different notions of curvature that are defined on graphs, namely, combinatorial curvature, Bakry-Émery curvature, and Ollivier's Ricci curvature. For each curvature notion, the definition and its motivation from Riemannian geometry will be explained. Moreover, we bring together some glob…

We examine the difference between several notions of curvature homogeneity and show that the notions introduced by Kowalski and Vanžurová are genuine generalizations of the ordinary notion of $k$-curvature homogeneity. The homothety group plays an essential role in the analysis.

Paper shows equivalence of two curvature notions on singular surfaces.

The study examines discrete curvature notions on Cayley graphs of certain groups.

Timelike curves in Lorentzian length spaces have a total curvature notion that agrees with smooth curves.

We study different notions of Riemannian curvatures: The $p$-curvatures which interpolate between the scalar curvature and the sectional curvature, the Gauss-Bonnet-Weyl curvatures form another interpolation from the scalar curvature to the Gauss-Bonnet integrand. We bring out the $(p,q)$-curvatures, which incorporate …

Here a new notion of fractional length of a smooth curve, which depends on a parameter $σ$, is introduced that is analogous to the fractional perimeter functional of sets that has been studied in recent years. It is shown that in an appropriate limit the fractional length converges to the traditional notion of length u…

Surveying problems with positive curvature forms, focusing on Ricci flow.

Curvature criteria for A-simple singularities and their parallel curves identified.

We examine the difference between several notions of curvature homogeneity and show that the notions introduced by Kowalski and Vanzurova are genuine generalizations of the ordinary notion of k-curvature homogeneity. The homothety group plays an essential role in the analysis. We give a complete classification of homot…

Study on curvature in finitely generated groups, showing positive curvature in specific cases.

Defines projective Ricci curvature and proves rigidity for sprays.

We study the sectional curvature of plane distributions on 3-manifolds. We show that if the distribution is a contact structure it is easy to manipulate this curvature. As a corollary we obtain that for every transversally oriented contact structure on a closed 3-dimensional manifold $M$ there is a metric, such that th…

We study a new notion of Ricci curvature that applies to Markov chains on discrete spaces. This notion relies on geodesic convexity of the entropy and is analogous to the one introduced by Lott, Sturm, and Villani for geodesic measure spaces. In order to apply to the discrete setting, the role of the Wasserstein metric…

We review recent results on classifying complete constant mean curvature 1 (CMC 1) surfaces in hyperbolic 3-space with low total curvature. There are two natural notions of "total curvature" -- one is the total absolute curvature, which is the integral over the surface of the absolute value of the Gaussian curvature, a…

In [11], I. M. Gelfand, V. Retakh, and M. Shubin defined the symplectic sectional curvature of a torsion-free connection preserving a symplectic form. The present article defines the corresponding notion of constant symplectic sectional curvature and characterizes this notion in terms of the curvature tensor of the sym…

In this paper we introduce two new notions of sectional curvature for Riemannian manifolds with density. Under both notions of curvature we classify the constant curvature manifolds. We also prove generalizations of the theorems of Cartan-Hadamard, Synge, and Bonnet-Myers as well as a generalization of the (non-smooth)…

We survey our recent results on classifying complete constant mean curvature 1 (CMC-1) surfaces in hyperbolic 3-space with low total curvature. There are two natural notions of "total curvature"-- one is the total absolute curvature which is the integral over the surface of the absolute value of the Gaussian curvature,…

The study examines Ricci solitons and curvature inheritance on Robinson-Trautman spacetimes.

For surfaces without boundary, nonlocal notions of directional and mean curvatures have been recently given. Here, we develop alternative notions, special cases of which apply to surfaces with boundary. Our main tool is a new fractional or nonlocal area functional for compact surfaces.

We have performed an empirical comparison of two distinct notions of discrete Ricci curvature for graphs or networks, namely, the Forman-Ricci curvature and Ollivier-Ricci curvature. Importantly, these two discretizations of the Ricci curvature were developed based on different properties of the classical smooth notion…

In this paper we study the notion of geodesic curvature of smooth horizontal curves parametrized by arc lenght in the Heisenberg group, that is the simplest sub-Riemannian structure. Our goal is to give a metric interpretation of this notion of geodesic curvature as the first corrective term in the Taylor expansion of …

Study curvature of piecewise metrics using moving frames.

The Cheeger-Gromoll theorem is adapted for groups, revealing geometric properties of virtually abelian groups.

The paper studies curvature tensors and hypersurfaces in Kenmotsu type manifolds.

Defines curvature for metric triples in metric spaces.

We discuss notions of Gauss curvature and mean curvature for polyhedral surfaces. The discretizations are guided by the principle of preserving integral relations for curvatures, like the Gauss/Bonnet theorem and the mean-curvature force balance equation.

The paper examines geometric properties of a unique spacetime model.

The problem of defining correctly geometric objects such as the curvature is a hard one in discrete geometry. In 2009, Ollivier defined a notion of curvature applicable to a wide category of measured metric spaces, in particular to graphs. He named it coarse Ricci curvature because it coincides, up to some given factor…

We prove that in two dimensions the synthetic notions of lower bounds on sectional and on Ricci curvature coincide.

Equivalence proven between different Ricci curvature definitions.

In this paper, we find a condition on $(α, β)$-metrics under which the notions of isotropic S-curvature, weakly isotropic S-curvature and isotropic mean Berwald curvature are equivalent.

Study Bismut-Griffiths-positivity in non-Kähler manifolds under Hermitian curvature flows.

We present a survey of some results and questions related to the notion of scalar curvature in the setting of symplectic supermanifolds.

New sprays of constant curvature introduced; conditions for metrizability given.

We use a Riemannnian approximation scheme to define a notion of $\textit{sub-Riemannian Gaussian curvature}$ for a Euclidean $C^{2}$-smooth surface in the Heisenberg group $\mathbb{H}$ away from characteristic points, and a notion of $\textit{sub-Riemannian signed geodesic curvature}$ for Euclidean $C^{2}$-smooth curve…

Study curvature and torsion from cross-ratios in discrete curves.

Study on stability of constant mean curvature hypersurfaces in Riemannian manifolds.

Our understanding of the notion of curvature in a noncommutative setting has progressed substantially in the past ten years. This new episode in noncommutative geometry started when a Gauss-Bonnet theorem was proved by Connes and Tretkoff for a curved noncommutative two torus. Ideas from spectral geometry and heat kern…

The paper extends a theorem about Kähler manifolds with quasi-negative curvature to almost quasi-negative curvature.

Graph curvature measured by inverse resistance distance.

New theorems compare Laplacian on Kähler manifolds.

Intertwining curvature bounds for graphs and quantum Markov semigroups verified.

The paper generalizes radial curvature bounds on manifolds.

A new type of sectional curvature is introduced. The notion is purely algebraic and can be located in linear algebra as well as in differential geometry.

In this paper we introduce the notion of timelike surface with harmonic inverse mean curvature in 3-dimensional Lorentzian space forms, and study their fundamental properties.

The aim of this paper is to introduce a notion of mean curvature flow soliton general enough to encompass target spaces of constant sectional curvature, Riemannian products or, in increasing generality, warped product spaces.