Using the methods of moving frames and exterior differential systems, we show that there exist Hopf hypersurfaces in complex hyperbolic space CH with any specified value of the Hopf principal curvature α less than or equal to the corresponding value for the horosphere. We give a construction for all such hypersurfaces in terms of Weierstrass-type data, and also obtain a classification of pseudo...

In this paper we establish a general framework for so-called parametric, polynomial, interpolation methods for parametric curves. In contrast to traditional methods, which typically approximate the components of the curve separately, parametric methods utilize geometric information (which depends on all the components) about the curve to generate the interpolant. The general framework suggests ...

In this note, we announce some results showing unexpected similarities between the moduli spaces of constant curvature metrics on 2-manifolds (the Riemann moduli space) and moduli spaces of Einstein metrics on 4manifolds. Let J? denote the moduli space of Einstein metrics of volume 1 on a compact, orientable 4-manifold M. If J£\ denotes the space of smooth Riemannian metrics of volume 1 on M, e...

We derive firstand second-order necessary optimality conditions for set-constrained optimization problems under the constraint qualificationtype conditions significantly weaker than Robinson’s constraint qualification. Our development relies on the so-called 2-regularity concept, and unifies and extends the previous studies based on this concept. Specifically, in our setting constraints are giv...

There is growing interest in using the close connection between differential geometry and statistics to model smooth manifold-valued data. In particular, much work has been done recently to generalize principal component analysis (PCA), the method of dimension reduction in linear spaces, to Riemannian manifolds. One such generalization is known as principal geodesic analysis (PGA). This paper, ...

Here are studied qualitative properties of the families of curves –foliations– on a surface immersed in R 4 , along which it bends extremally in the direction of the mean normal curvature vector. Typical singularities and cycles are described, which provide sufficient conditions, likely to be also necessary, for the structural stability of the configuration of such foliations and their singular...

Anisotropic diffusion is known to be a powerful tool in image processing. It enables the smoothing of initially noisy images while still retaining, respectively sharpening edges and enhancing features. Here recent results in the context of vector field visualization are expanded to non Euclidean domains. The aim is to graphically represent vector field data on two dimensional surfaces in an int...

We study the motion of surfaces in an intrinsic formulation in which the surface is described by its metric and curvature tensors. The evolution equations for the six quantities contained in these tensors are reduced in number in two cases: (i) for arbitrary surfaces, we use principal coordinates to obtain two equations for the two principal curvatures, highlighting the similarity with the equa...

Curvature measure is one of the basic notion in the theory of convex bodies. Together with surface area measures, they play fundamental roles in the study of convex bodies. They are closely related to the differential geometry and integral geometry of convex hypersurfaces. Let Ω is a bounded convex body in R with C2 boundary M , the corresponding curvature measures and surface area measures of ...

A new method for noise removal of arbitrary surfaces meshes is presented which focuses on the preservation and sharpening of non-linear geometric features such as curved surface regions and feature lines. Our method uses a prescribed mean curvature flow (PMC) for simplicial surfaces which is based on three new contributions: 1. the definition and efficient calculation of a discrete shape operat...