Normal Gauss Map of a Tight Smooth Surface in (R sup 3)

摘要

The normal Gauss map of a tight smooth surface in R sup 3 is reminiscent ofminimal surfaces and holomorphic functions. The outside unit normal vectors of a convex smooth (of class C(sup infinity symbol)) surface M included in R sup 3 determine the Gauss map g : M -> S(sup 2) into the unit sphere. For a strictly convex surface, respectively an analytic convex surface, g is a diffeomorphism, resp. a homeomorphism. But in general it is a cell-like map, namely, for which the inverse image g(sup -1)(z) for z a member of S(sup 2) is cellular because convex. It is the intersection of a nested sequence of open discs in M. The authors will generalize this to all tight surfaces in theorems 1, 2, 3. Analogous theorems hold for locally tight surfaces.