Tight surfaces in \(E^3\): structure theorems (Q1298183)

This interesting paper analyzes the structure of smooth tight immersions of surfaces into Euclidean 3-space. The key observation is that the Gauss map into the 2-sphere is essentially a branched covering. The induced stratification of the surface plays an important role. The main part of the results was previously announced by the author and \textit{N. H. Kuiper} [in: Chern -- a great geometer, 161-175 (1952; Zbl 0828.53003)]. Theorem A states that for every smooth tight immersion, the Gauss map admits a factorization into a cellular map followed by a branched covering. Theorem B gives upper and lower bounds for the number of Morse-critical sets of height functions, i.e., those critical sets where more than just one \(k\)-cell has to be attached when passing through the critical level. Particular cases of Morse-singular critical sets are the so-called essential top-sets. Theorem D describes the decomposition into the various parts of positive and negative curvature. Theorems E and F deal with tight immersions in non-standard regular homotopy classes. In particular, it is shown that an orientable surface of genus 2 does not admit any tight immersion which is not regularly homotopic to the standard embedding. This was one of the open cases. Several other cases are still open in which there are polyhedral tight immersions [see \textit{D. P. Cervone}, Topology 35, 863-873 (1996; Zbl 0858.53051)].