Tightness and Polar Height Functions:

It is often convenient to consider tightness in terms of height functions: given an immersion of a surface and a direction z in space, the height function in this direction is simply the projection of the immersed surface onto the directed line in the direction of z. For almost all directions, the critical points of a height function are isolated; the height function in such a direction is called a Morse height function.

For smooth surfaces, a Morse height function will have critical points where the tangent planes to the surface are perpendicular to the direction z, namely local minima, local maxima and saddles. Polyhedral surfaces will have critical points at the corresponding piecewise linear structures.

The curvature at a maximum or a minimum is positive, and since a tight surface has all its positive curvature on its convex envelope , a Morse height function on a tight immersion will have exactly one maximum and one minimum. Furthermore, since the Euler characteristic is equal to the number of maxima plus the number of minima minus the number of saddle points, such a height function will have the fewest possible number of saddles. A Morse height function with this property is called a polar height function.

This observation leads to the following characterization of tight surfaces:

Theorem: An immersion of a closed, compact, connected surface is tight if, and only if, every Morse height function on it is polar.

Since minima and maxima are exchanged by reversing the direction, and since the number of maxima and minima determine the number of saddles, we have the following

Corollary: An immersion of a closed, compact, connected surface is tight if, and only if, every Morse height function on it has exactly one local maximum.

Tightness and the two-piece property

Tightness and the convex hull

Tightness and its consequences

Kuiper's initial question

8/8/94 dpvc@geom.umn.edu -- The Geometry Center