Related papers: Ricci Curvature and the Mechanics of Solids

The continuum mechanics of line defects representing singularities due to terminating discontinuities of the elastic displacement and its gradient field is developed. The development is intended for application to coupled phase…

Giving explicit parametrizations of discrete constant Gaussian curvature surfaces of revolution that are defined from an integrable systems approach, we study Ricci flow for discrete surfaces, and see how discrete surfaces of revolution…

We discuss the geometry of warped foliations. After examining the Levi-Civita connection, we describe the formulae for sectional, Ricci and scalar curvatures. In the final part of this note, we present some examples.

Final version in paper linked above.

In the article, we exhibit a series of new examples of rigid plane curves, that is, curves, whose collection of singularities determines them almost uniquely up to a projective transformation of the plane.

We discuss in which sense general metric measure spaces possess a first order differential structure. Building on this, we then see that on spaces with Ricci curvature bounded from below a second order calculus can be developed, permitting…

In this survey paper, we analyse and compare the recent curvature estimates for three types of $4$-dimensional gradient Ricci solitons, especially between Ricci shrinkers [58] and expanders [17]. In addition, we provide some new curvature…

In this paper, we derive certain curvature estimates for 4-dimensional gradient steady Ricci solitons either with positive Ricci curvature or with scalar curvature decay.

In the paper we discuss gap phenomena of three different types related to Ricci (and sectional) curvature. The first type is about spectral gaps. The second type is about sharp gap metric-rigidity, originally due to Anderson. The third is…

We consider an inverse variational problem for the lines of constant curvature in (pseudo-)Euclidean two-, three-, and four-dimensional spaces. The accumulated results are physically meaningful in the case of relativistic mechanics of…

We briefly review a number of important recent experimental and theoretical developments in the field of dynamic fracture. Topics include experimental validation of the equations of motion for straight tensile cracks (in both infinite media…

In a way similar to the continuous case formally, we define in different but equivalent manners the difference discrete connection and curvature on discrete vector bundle over the regular lattice as base space. We deal with the difference…

We survey work of Lott-Villani and Sturm on lower Ricci curvature bounds for metric-measure spaces.

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.…

There is described a spacetime formulation of both nonrelativistic and relativistic elasticity. Specific attention is devoted to the causal structure of the theories and the availability of local existence theorems for the initial-value…

This article introduces a new approach to discrete curvature based on the concept of effective resistances. We propose a curvature on the nodes and links of a graph and present the evidence for their interpretation as a curvature. Notably,…

In this paper, we prove some rigidity theorems for shrinking gradient Ricci solitons with nonnegative sectional curvature.

Gradient steady Ricci solitons are natural generalizations of Ricci-flat manifolds. In this article, we prove a curvature gap theorem for gradient steady Ricci solitons with nonconstant potential functions; and a curvature gap theorem for…

In this short note we review some known results on the structure and regularity of spaces with lower Ricci curvature bounds. We present some known and new open questions about next steps.

We study integral and pointwise bounds on the curvature of gradient shrinking Ricci solitons. As applications we discuss gap and compactness results for gradient shrinkers.