Tangential and normal Euler numbers, complex points, and singularities of projections for oriented surfaces in four-space
type: Item
instance of: scholarly article
title: Tangential and normal Euler numbers, complex points, and singularities of projections for oriented surfaces in four-space
zbMATH Open document ID: 0815.57024
published in: Pacific Journal of Mathematics
publication date: 1995-07-09T00:00:00Z
review text: Given a generic smooth immersion \(X\) of a closed oriented surface \(M\) into \(\mathbb{R}^ 4\), one can count (i) the zeroes of a generic tangential vector field along \(M\), (ii) the zeroes of a generic normal vector field, and (iii) the complex points of \(X\) (i.e. the points of \(M\) where the Gauss map of \(X\) yields a complex line in \(\mathbb{R}^ 4 = \mathbb{C}^ 2)\). with \(\pm\) signs in a natural way. The authors present a concrete geometric proof of the theorem that the sum of the resulting integers vanishes. This is based on a detailed study of the Grassmann manifold \(G(2,4)\) of oriented 2-planes in \(\mathbb{R}^ 4\). They produce a (partial) 3-dimensional bordism in \(G(2,4)\) which relates the three 2-dimensional submanifolds \(T,N\) and \(C\) whose intersections with the Gauss map of \(X\) yield the three sets (i), (ii) and (iii) above. The various concepts are carefully explained and illustrated by a number of concrete examples and computer figures.
Mathematics Subject Classification ID: 57R42
zbMATH DE Number: 540675
zbMATH Keywords: generic smooth immersion of a closed oriented surface into \(\mathbb{R}^ 4\)
zbMATH Keywords: zeroes of a generic tangential vector
zbMATH Keywords: zeroes of a generic normal vector field
zbMATH Keywords: complex points
zbMATH Keywords: Gauss map
author: Tom Banchoff
author: Frank A. Farris
reviewed by: Ulrich Koschorke
MaRDI profile type: Publication
full work available at URL: pjm.1993.161.1
OpenAlex ID: W1975379108
DOI: 10.2140/PJM.1993.161.1
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description: scientific article; zbMATH DE number 540675