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.*

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

*The Geometry Center*