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Tight vs Taut
Posted:
Nov 3, 1993 3:42 PM


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 nondegenerate Euclidean distance function L_p has the minimum number of critical points. For n=2, this is equivalent to the spherical twopiece 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 3sphere). Tautness is a stronger condition than tightness. For example, a taut nsphere must me a round sphere, not merely convex".
[1] T.F. Banchoff, The spherical twopiece property and tight surfaces in spheres, J. Diff. Geom. 1 (1967) 245256.
Banchoff's spherical twopiece property is similar to the twopiece property I mentioned previously, except this time the cutting is by spheres rather than by planes. That is, a surface has the spherical twopiece 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 3sphere, 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 3sphere. 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 4space 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 selfintersection (these correspond to projections from points not on the 3sphere). 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 twopiece property with the definition above.
There is a corresponding idea for the (planar) twopiece property, namely, that a surface has the twopiece 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 "topset", and when that set has nontrivial homology, as in the torus example, it is called a "topcycle" (this is not stated quite properly, as the topset may not be a cycle, but the idea can be made precise, and this is the right intuition). The number and shape of topcycles 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



