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Topic: Tight vs Taut
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Davide P. Cervone

Posts: 4
Registered: 12/6/04
Tight vs Taut
Posted: Nov 3, 1993 3:42 PM
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Several people have sent me mail asking what a "taut" immersions is
(as opposed to "tight"). Taking the definition from Cecil and Ryan's
book, Tight and taut immersions of manifolds, Research Notes in Math.
107, Pitman Publ., 1985:

"A smooth immersion f:M^n->E^m is taut if every non-degenerate
Euclidean distance function L_p has the minimum number of critical
points. For n=2, this is equivalent to the spherical two-piece
property of Banchoff [1] who showed that a taut compact surface in E^3
must be a round sphere or a cyclide of Dupin (image under
stereographic projection of a product torus in the 3-sphere).
Tautness is a stronger condition than tightness. For example, a taut
n-sphere must me a round sphere, not merely convex".

[1] T.F. Banchoff, The spherical two-piece property and tight surfaces
in spheres, J. Diff. Geom. 1 (1967) 245--256.

Banchoff's spherical two-piece property is similar to the two-piece
property I mentioned previously, except this time the cutting is by
spheres rather than by planes. That is, a surface has the spherical
two-piece property if every sphere cuts it into at most two parts.

For those of you who haven't seen the cyclides of Dupin, you can
generate them by taking a torus on the 3-sphere, such as the one given
by (cos u, sin u, cos v, sin v) for 0 <= u < 2pi, 0 <= v < 2pi, and
projecting it stereographically from different points on the 3-sphere.
The projection from the north pole (0,0,0,1) is just the standard
torus of revolution, but projections from other points (or
equivalently, the projection from the north pole of a rotation in
4-space of this torus) give cyclides of Dupin. If memory serves me,
the cyclides have the nice property that their offset surfaces are
again cyclides, though they may include self-intersection (these
correspond to projections from points not on the 3-sphere). For
example, think of offsets to the standard torus of revolution where
the offsets meet or pass through each other in the center.

The distance functions mentioned by Cecil and Ryan are Euclidean
distances from a fixed point. They represent, in some sense, all the
spheres centered at p. The fact that they have the minimum number of
critical points means that they have, in general, only one maximum and
one minimum, and the rest of the critical points are saddles (at least
for n=2 and m=3). If some sphere centered at p cut the surface into
three pieces, then if two pieces are inside each must contain a
minimum, while if two are outside, each must contain a maximum for the
distance function from p. This won't happen if L_p has only one of
each. Conversely, if there are two minima (say), then a sphere or
radius just larger than the farther of the two minima will cut the
surface into three parts (one containing a small neighborhood of the
farther minimum, one containing the other minimum, and the remainder
of the surface outside the sphere). This indicates the equivalence of
the spherical two-piece property with the definition above.

There is a corresponding idea for the (planar) two-piece property,
namely, that a surface has the two-piece property if, and only if,
every Morse height function is polar (i.e., has exactly one maximum
and one minimum; that is, has the minimum number of critical points).
Here, a height function is the projection of the surface onto a
directed line though the origin, and Morse means essentially that the
critical points are isolated. An argument similar to the one above
can be used to prove this.

The study of height functions is central to an understanding of tight
surfaces, particularly those height functions that are not Morse. For
example, consider the torus of revolution, and the height function in
the direction of its axis of revolution. The maximum and minimum
values occur not at a single point, but along a circle (the circle
where the torus would sit if you placed it on a table, and where a
piece of cardboard would touch it if placed on top of the torus). The
sets one which a height function achieves its absolute maximum is
called a "top-set", and when that set has non-trivial homology, as in
the torus example, it is called a "top-cycle" (this is not stated
quite properly, as the top-set may not be a cycle, but the idea can be
made precise, and this is the right intuition). The number and shape
of top-cycles is very important to the study of tight immersions, and
it is in fact crucial to Haab's proof that the real projective plane
with one handle has no smooth tight immersion.

Davide P. Cervone

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