Recognizing the 3-sphere

Recognizing Phosphatidylinositol 3-Phosphate

Phosphatidylinositol 3-phosphate directs the endosomal localization of regulatory proteins by binding to FYVE and PX domains. New structures of these domains complexed with the phosphoinositide headgroup show how interactions with phosphate and hydroxyl groups differentiate this lipid from all others.

On Involutions of the 3-Sphere

Spherical Minimal Immersions of the 3-sphere

In 1966 Takahashi [11] proved that a minimal isometric immersion f : S(1) → S(r) of round spheres exists iff r = √ m/λp, where λp is the pth eigenvalue of the Laplacian on S; in this case, the components of f are spherical harmonics on S of order p. This immersion is unique up to congruence on the range and agrees with the generalized Veronese map if m = 2 as was shown in 1967 by Calabi [1]. In...

Bonnet Pairs in the 3 - Sphere

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...

Sphere packings in 3-space

In this paper we survey results on packings of congruent spheres in 3-dimensional spaces of constant curvature. The topics discussed are as follows: Hadwiger numbers of convex bodies and kissing numbers of spheres; Touching numbers of convex bodies; Newton numbers of convex bodies; One-sided Hadwiger and kissing numbers; Contact graphs of finite packings and the combinatorial Kepler problem; Is...