Tight immersions of surfaces in \(E^ 3\)
type: Item
instance of: scholarly article
title: Tight immersions of surfaces in \(E^ 3\)
zbMATH Open document ID: 0763.53060
DOI: 10.1007/BF02566495
author: François Haab
published in: Commentarii Mathematici Helvetici
publication date: 1993-01-16T00:00:00Z
full work available at URL: https://eudml.org/doc/140264
review text: This paper solves a long-standing problem posed by \textit{N. H. Kuiper} in Comment. Math. Helv. 35, 85-92 (1961; Zbl 0243.53043): Does there exist a tight immersion of the projective plane with one handle into \(E^ 3\)? In other words: Does there exist an immersion of this surface with total absolute curvature \(\int| K|=2\pi\cdot 5\)? The main result (Théorème A) says that there is no smooth immersion of this kind. Besides standard techniques like critical point theory, top-cycles, singularities of excellent (2-generic) mappings, the proof involves a new idea developed by the author. This concerns the global structure of the profile curves associated with linear projections of the surface to the plane, namely the part of the surface which is folded by such a projection. Assuming the existence of a tight surface with \(\chi=-1\), a careful combination of these methods leads to a contradiction.
reviewed by: Wolfgang Kühnel
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 57R45
Mathematics Subject Classification ID: 57R42
zbMATH DE Number: 93803
zbMATH Keywords: Gauss mapping
zbMATH Keywords: Boy surface
zbMATH Keywords: projective plane
zbMATH Keywords: total absolute curvature
zbMATH Keywords: critical point theory
zbMATH Keywords: top-cycles
zbMATH Keywords: profile curves
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 93803