Tautness and Lie sphere geometry (Q1097527)

This paper is essential reading for everyone interested in tight and taut immersions. The main result that it contains is the invariance of tautness for immersions under the transformation group of the Lie geometry of spheres. This group is considerably larger than the Moebius group, under which tautness is well-known to be invariant. Thus Lie sphere geometry will play an important role in the further development of taut immersions. The paper begins with a careful presentation of Lie sphere geometry and its transformation group. It then defines the Legendre map of an immersion. Finally, it relates all this to tautness.