Tightness and the Two-Piece Property:

At first glance, this may seem unrelated to tightness, which was originally defined in terms of the total absolute curvature integral, but it turns out that the two ideas are equivalent. To see this, we use the fact that an immersion is tight if and only if every Morse height function has exactly one local maximum.

First, suppose that an immersion has the two-piece property, and consider a direction for which the associated height function is Morse. Suppose there are two local maxima for this function. Then a plane just below the smaller of the two maxima cuts a small neighborhood of the maximum away from the rest of the surface. In addition, it also cuts off a larger, but disjoint, patch that contains the other maximum. The remainder of the surface below the plane makes a third piece, contradicting the fact that the surface has the two-piece property. Thus there can be only one maximum for any Morse height function, and the surface is tight.

Conversely, suppose a tight immersion is cut by a plane, and consider the height function in the direction of the normal to this plane. (We can assume this is a Morse height function, for if not, since almost all height functions are Morse, a slight tilting of the plane will yield such a function without changing the number of components into which the plane cuts the surface.) Now each component of the surface that lies above the plane must contain a local maximum for the height function; but since the immersion is tight, every Morse height function has a single local maximum, so there is only one component above the plane. Similarly, by considering the height function in the opposite direction, there can be only a single component below the plane, so the plane divides the surface into exactly two pieces. This holds for all the planes that divide the surface, so the immersion has the two-piece property.

Thus we have the following theorem originally due to Thomas Banchoff [B1].

Theorem: An immersion of a closed, compact, connected surface is tight if, and only if, it has the two-piece property.

This theorem gives a concrete geometric interpretation of tightness, and provides an easily visualized criterion for determining whether or not a surface is tightly immersed. Furthermore, this property makes sense for both smooth and polyhedral surfaces, with or without boundary, whereas the definition in terms of the total absolute curvature integral relied on smoothness and the fact that the surface is closed.