Since the integrand is always positive, the only way this integral can be zero is for the region of integration to be empty. This means that all the positive curvature must be on the convex envelope of the surface, an observation that leads to the following important geometric description of tight surfaces:

**Theorem:** *An immersion of a
closed,
compact,
connected
surface M is tight if, and only if, it can be decomposed into two
components, M+ and M-, such that*

*The curvature is non-negative on**M+*and non-positive on*M-*,*The image of**M+*is an embedding and is equal to the convex envelope of the image of*M*minus a finite number of planar, convex disks,*The boundary of each of these disks is the image of a non-trivial 1-cycle in**M.*These curves are called*top cycles*.

For example, the torus of revolution is tight (below left).
Its decomposition into the *M+* and *M-* regions is shown
(right). Notice the two top cycles that form the common boundary of the
*M+* and *M-* regions.

* 8/8/94 dpvc@geom.umn.edu -- *

*The Geometry Center*