Introduction

The theory of tight submanifolds starts with attempts to generalize theorems about convex surfaces to topologically more complex surfaces such as the torus. For surfaces, it is possible to develop this generalization in terms of an elementary notion, the two-piece property, which then leads to the study of critical points of height functions and the theory of total absolute curvature. These notions can then be applied for higher-dimensional objects in higher-dimensional Euclidean spaces, producing a rich collection of examples and theorems in the global geometry of submanifolds.

An object in ordinary three-dimensional space is said to have the two-piece property, or TPP, if any plane cuts it into at most two pieces. Examples of surfaces with the TPP are spheres and ellipsoids and, more generally, the boundary of any bounded convex body. There are also nonconvex objects with boundaries that have the TPP: for example, a torus of revolution (Figure 1), or, more generally, a surface of revolution obtained by revolving a convex curve around an axis in the plane of the curve and not meeting the curve.