Recall that the radius of a compact metric space (X, dist) is given by rad X = minx∈X maxy∈X dist(x, y). In this paper we generalize Berger’s 1 4 -pinched rigidity theorem and show that a closed, simply connected, Riemannian manifold with sectional curvature ≥ 1 and radius ≥ π2 is either homeomorphic to the sphere or isometric to a compact rank one symmetric space. The classical sphere theorem ...

A practical example of B-spline curve control points manipulation for the geometric construction of a free form shape is presented. Elements of a cross-sectional design methodology are used in conjunction with a skinning type operator for the definition of a B-spline surface. Skinning process are well established in the CAD community but further difficulties arise in producing smooth surfaces u...

A metric complex M is a connected, locally-finite simplicial complex linearly embedded in some Euclidean space R. Metric complexes M and M' are isometric if they have subdivisions L and L and if there is a simplicial isomorphism h:L -• L such that for every a e L, the linear map determined by h\a -• h(a) is an isometry (that is, it extends to an isometry of the affine spaces generated by these ...

It is fair to say that Riemannian geometry started with Gauss’s famous ”Disquisitiones generales” from 1827 in which one finds a rigorous discussion of what we now call the Gauss curvature of a surface. Much has been written about the importance and influence of this paper, see in particular the article [Do] by P.Dombrowski for a careful discussion of its contents and influence during that time...

I. Introduction-a quick historical survey of geodesic flows on negatively curved spaces. II. Preliminaries on Riemannian manifolds A. Riemannian metric and Riemannian volume element B. Levi Civita connection and covariant differentiation along curves C. Parallel translation of vectors along curves D. Curvature E. Geodesics and geodesic flow F. Riemannian exponential map and Jacobi vector fields...

On a 3-dimensional Lorentzian manifold, the sectional curvature function at any point may be represented as a rational function (quotient of two quadratics) on the real projective plane [2]. (More generally, the sectional curvature at a point of any pseudoRiemannian n-manifold may be represented as a rational function on the Grassmann variety of planes in n-space.) Using this representation, Be...

As a means to better understanding manifolds with positive curvature, there has been much recent interest in the study of nonnegatively curved manifolds which contain either a point or an open dense set of points at which all 2-planes have positive curvature. We study infinite families of biquotients defined by Eschenburg and Bazaikin from this viewpoint, together with torus quotients of S × S....

Together with spaces of constant sectional curvature and products of a real line with a manifold of constant curvature, the socalled Egorov spaces and ε-spaces exhaust the class of n-dimensional Lorentzian manifolds admitting a group of isometries of dimension at least 1 2n(n − 1) + 1, for almost all values of n [Patrangenaru V., Geom. Dedicata 102 (2003), 25–33]. We shall prove that the curvat...

This announcement describes research concerning local quasiconvexity, coherence, compact cores, and local indicability for fundamental groups of certain 2-complexes.

Let (M,g) be a compact oriented 4-dimensional Einstein manifold. If M has positive intersection form and g has non-negative sectional curvature, we show that, up to rescaling and isometry, (M,g) is CP2, with its standard Fubini-Study metric.