Tangential and normal Euler numbers, complex points, and singularities of projections for oriented surfaces in four-space
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.
generic smooth immersion of a closed oriented surface into \(\mathbb{R}^ 4\)
scientific article; zbMATH DE number 540675