Tautness and Lie sphere geometry

Abstract

An immersion f of a smooth compact connected manifold M into Euclidean space E" is said to be taut if every Morse function of the form Lp(x)= Ip--f(x)l 2, p ~ lR", has the minimum number of critical points required by the Morse inequalities on M. Carter and West [3] showed that tautness is invariant under Moebius (conformal) transformations of IR" and under stereographic projection from IR" into the unit sphere S" in IR" + 1. They showed further that a taut immersion must, in fact, be an embedding. Pinkall [13] showed that for the theory of taut embeddings, it is sufficient to study hypersurfaces in the following sense. Let M (11" (or S") be a compact submanifold of codimension greater than one, and let M s be a tube of sufficiently small radius ~. about M so that M~ is an embedded hypersurface. Then M is taut if and only if M s is taut. Cecil and Ryan [6] showed that any isoparametric hypersurface in S" is taut, and recently, Hsiang, Palais and Terng 1-8] have proven that any isoparametric submanifold of any codimension is taut. The main result of this paper is that within the class of immersions, tautness is invariant under the group of Lie sphere transformations. This group is considerably larger than the group of Moebius transformations, as we will now explain. In his work on contact transformations, Lie 1-9] developed his geometry of oriented spheres (see also Blaschke [2]). Lie established a bijective correspondence between the set of all oriented hyperspheres and point spheres in S" and the points on the quadric hypersurface Q,+I in real projective space pn+2 given by the equation (x, x ) = 0, where ( , ) is an indefinite metric with signature (n + 1,2) on N "+3. Q"+~ contains projective lines but no linear subspaces of pn+2 of higher dimension. The 1-parameter family of oriented spheres, called a parabolic pencil, corresponding to the points of a projective line lying on Q,+l consists of all oriented hyperspheres which are in oriented contact at a certain contact element on S". In this way, Lie established a diffeomorphism between the manifold of contact elements on S" and the space A 2"1 of projective lines which lie on Q"+ 1. A