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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Science+Business Media New York
About this chapter
Cite this chapter
Petersen, P. (1998). Hypersurfaces. In: Riemannian Geometry. Graduate Texts in Mathematics, vol 171. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-6434-5_4
Download citation
DOI: https://doi.org/10.1007/978-1-4757-6434-5_4
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4757-6436-9
Online ISBN: 978-1-4757-6434-5
eBook Packages: Springer Book Archive