Related papers: How Riemannian Manifolds Converge: A Survey

Random geometric graphs are random graph models defined on metric measure spaces. A random geometric graph is generated by first sampling points from a metric space and then connecting each pair of sampled points independently with a…

We deal with irregular curves contained in smooth, closed, and compact surfaces. For curves with finite total intrinsic curvature, a weak notion of parallel transport of tangent vector fields is well-defined in the Sobolev setting. Also,…

We review the notion of shape tensor of an embedded manifold, which efficiently combines intrinsic and extrinsic geometry, and allows for intuitive understanding of some basic concepts of classical differential geometry, such as parallel…

Among plenty of applications, low-dimensional homogeneous spaces appear in cosmological models as both, classical factor spaces of multidimensional geometry and minisuperspaces in canonical quantization. Here a new tool to restrict their…

We establish a one-to-one correspondence between static spacetimes and Riemannian manifolds that maps causal geodesics to geodesics, as suggested by L. C. Epstein. We then explore constant curvature spacetimes - such as the de Sitter and…

This paper considers the problem of minimizing the summation of a differentiable function and a nonsmooth function on a Riemannian manifold. In recent years, proximal gradient method and its invariants have been generalized to the…

This paper is a survey of some of the developments in coarse extrinsic geometry since its inception in the work of Gromov. Distortion, as measured by comparing the diameter of balls relative to different metrics, can be regarded as one of…

In this paper, we investigate the geometry of Einstein-type equation on a Riemannian manifold, unifying various particular geometric structures recently studied in the literature, such as critical point equation and vacuum static equation.…

We discuss domestic affairs of metric spaces, keeping away from any extra structure. Topics include universal spaces, injective spaces, Hausdorff and Gromov--Hausdorff convergences, and ultralimits.

Let $M$ be a compact Riemannian manifold not containing any totally geodesic surface. Our main result shows that then the area of any complete surface immersed into $M$ is bounded by a multiple of its extrinsic curvature energy, i.e. by a…

Manifold reconstruction has been extensively studied for the last decade or so, especially in two and three dimensions. Recently, significant improvements were made in higher dimensions, leading to new methods to reconstruct large classes…

The curvature discussed in this paper is a rather far going generalization of the Riemannian sectional curvature. We define it for a wide class of optimal control problems: a unified framework including geometric structures such as…

In image set classification, a considerable advance has been made by modeling the original image sets by second order statistics or linear subspace, which typically lie on the Riemannian manifold. Specifically, they are Symmetric Positive…

For seven-dimensional Riemannian manifolds equipped with a $G_2$-structure, we show in a full detailed way that all integral formulas and divergence equations, given by diverse authors, are agree with the ones displayed here in terms of the…

We study the possibility that the universe is subjected to a deformation, besides its expansion described by Friedmann's equations. The concept of smooth deformation of a riemannian manifolds associated with the extrinsic curvature is…

A survey of some recent and important results which have to do with integrable equations and their relationship with the theory of surfaces is given. Some new results are also presented. The concept of the moving frame is examined, and it…

We investigate isometric immersions $f\colon M^n\to\R^{n+2}$, $n\geq 3$, of Riemannian manifolds into Euclidean space with codimension two that admit isometric deformations that preserve the metric of the Gauss map. In precise terms, the…

In Euclidean space, the integration by parts formula for a set of finite perimeter is expressed by the integration with respect to a type of surface measure. According to geometric measure theory, this surface measure is realized by the…

This chapter reviews some past and recent developments in shape comparison and analysis of curves based on the computation of intrinsic Riemannian metrics on the space of curves modulo shape-preserving transformations. We summarize the…

In this paper we extend to non-compact Riemannian manifolds with boundary the use of two important tools in the geometric analysis of compact spaces, namely, the weak maximum principle for subharmonic functions and the integration by parts.…