Our interest in horoballs comes on the one hand from the problem of describing the dynamics of certain important classes of self-maps of convex cones, and on the other hand from the work of Rieffel [17] on quantum metric spaces in the context of Connes’ noncommutative geometry. Horofunctions also appeared recently in a general law of large numbers for random walks on groups [11]. As is remarked...

We study the self force acting on a particle endowed with scalar charge, which is held static (with respect to an undragged, static observer at infinity) outside a stationary, axially-symmetric black hole. We find that the acceleration due to the self force is in the same direction as the black hole’s spin, and diverges when the particle approaches the outer boundary of the black hole’s ergosph...

This paper surveys aspects of the convergence and degeneration of Riemannian metrics on a given manifold M , and some recent applications of this theory to general relativity. The basic point of view of convergence/degeneration described here originates in the work of Gromov, cf. [27]-[29], with important prior work of Cheeger [14], leading to the joint work of [16]. This Cheeger-Gromov theory ...

In many areas of computer vision, such as multiscale analysis and shape description, an image or surface is smoothed by a nonlinear parabolic partial differential equation to eliminate noise and to reveal the large global features. An ideal flow, or smoothing process, should not create new features. In this paper we describe in detail the effect of a number of flows on surfaces on the parabolic...

A century ago, Salkowski [9] introduced a family of curves with constant curvature but non-constant torsion (Salkowski curves) and a family of curves with constant torsion but nonconstant curvature (anti-Salkowski curves). In this paper, we adapt definition of such curves to spacelike curves in Minkowski 3-space. Thereafter, we introduce an explicit parametrization of a spacelike Salkowski curv...

In this paper, we study n(n ≥ 3)-dimensional complete connected and oriented space-like hypersurfaces Mn in an (n+1)-dimensional Lorentzian space form Mn+1 1 (c) with non-zero constant k-th (k < n) mean curvature and two distinct principal curvatures λ and μ. We give some characterizations of Riemannian product H(c1) ×Mn−m(c2) and show that the Riemannian product Hm(c1)×Mn−m(c2) is the only com...

In this paper, a new ear anatomy feature edge extraction method based on Hessian matrix is proposed. Stable edge is obtained from principal curvature image across scale space. Firstly, the side face image that includes an ear is filtered and forms Gaussian pyramid. Secondly, the 2D gray image in the pyramid was regarded as a surface, maximum and minimum principal curvature and their direction w...

In the previous lecture we developed the notion of the curvature of a surface, showing a bunch of nice theoretical properties. But aside from theoretical elegance, what’s so great about curvature? Well, remember from the plane curve setting that the curvature function uniquely defines a curve parameterized by arc length up to a rigid motion. A similar result holds for surfaces, though the proof...

Two non-congruent surfaces that are isometric and have the same mean curvature at corresponding points are called a Bonnet pair of surfaces or simply a Bonnet pair. The interest in such pairs arises from the following considerations: One of the fundamental problems of surface theory is to find invariants which characterize surfaces geometrically. The Bonnet theorem states that a surface is dete...