A Gauss map and hybrid degree formula for compact hypersurfaces in Minkowski space
✍ Scribed by Marek Kossowski
- Publisher
- Springer
- Year
- 1989
- Tongue
- English
- Weight
- 546 KB
- Volume
- 32
- Category
- Article
- ISSN
- 0046-5755
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✦ Synopsis
A compact oriented hypersurface in four-dimensional Minkowski space contains a locus where the induced metric is degenerate. We show that if certain transversality conditions are satisfied then the degree of the hypersurface Gauss map is determined by the Euler characteristic of the degeneracy locus. This leads to a total curvature inequality with lower bound determind by the Betti numbers of the hypersurface. Equality characterizes particularly simple hypersurfaces.
0. Introduction
Every compact oriented hypersurface in four-dimensional Minkowski space contains a subset ~o where the induced metric degenerates. The objective of this paper is to construct a gauss map for such a hypersurface and to show that if certain transversality conditions are satisfied then the degree of the gauss map can be expressed in terms of the intrinsic differential topology of D °. Such a degree formula mixes the essential features of the Hopf-Poincar6 and Gauss-Bonnet formulae.
An outline of the paper is as follows. We begin by defining the gauss map (which is essentially a classifying map for the co-orthogonal bundle of a hypersurfaee). This is followed by a discussion of the relevant transversality conditions and their effect on the locus of metric degeneracy. We then prove the degree formula. Finally this formula and the ideas of [-5] lead us to a weak notion of total curvature. The author would like to thank R.