Convexity and cylindrical two-piece properties (Q1060439)

The authors continue to study k-cylindrical tautness and the k- cylindrical weak resp. strong two-piece property (k-cylindrical WTPP resp. STPP) for smooth immersions \(f: M\to R^ n\) of closed manifolds into euclidean n-space which were introduced in their joint paper with \textit{N. G. Mansour} [Math. Ann. 261, 133-139 (1982; Zbl 0492.53041)]. After some preliminary considerations for convex sets and top-sets they establish the following characterizations in the codimension 1 case: If f has the 2-cylindrical STPP and \(n\geq 4\), then M is diffeomorphic to \(S^{n-1}\) and f is a tight embedding. If f satisfies the stronger property to be 2-cylindrically taut, then the image of f is a round hypersphere. If \(M=S^{n-1}\) and f is k-cylindrically taut (k\(\leq n-2)\), then \(f(S^{n-1})\) is a round hypersphere too. Furthermore there are no 1-cylindrically taut immersions in higher codimensions. Up to now no non- trivial k-cylindrically taut immersions are known for \(k\neq 0\), n-1.