Hypersurfaces

Abstract

In this chapter we shall explain some of the classical results for hypersurfaces in Euclidean space. First we introduce the Gauss map and show that convex im­mersions are embeddings of spheres. We then establish a connection between convexity and positivity of the intrinsic curvatures. This connection will enable us to see that ℂP ² and the Berger spheres are not even locally hypersurfaces in Euclidean space. We give a brief description of some classical existence results for isometric embeddings. Finally, a description of the Gauss-Bonnet theorem and its generalizations is given. One thing one might hope to get out of this chapter is the feeling that positively curved objects somehow behave like convex hypersurfaces, and might therefore have a very restricted topological type.