• Peter Petersen
Part of the Graduate Texts in Mathematics book series (GTM, volume 171)


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.


Riemannian Manifold Curvature Operator Shape Operator Isometric Embedding Gauss Equation 
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Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Peter Petersen
    • 1
  1. 1.Department of MathematicsUniversity of California, Los AngelesLos AngelesUSA

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