Sectional Curvature Comparison I

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


We shall first classify spaces with constant curvature. The real subject of this chapter is how one can compare manifolds to spaces with constant curvature. We shall for instance prove the Hadamard-Cartan theorem, which says that a simply connected manifold with sec ≤ 0 is diffeomorphic to ℝ n . There are also some interesting restrictions on the topology in positive curvature that we shall inves­tigate, notably, Synge’s theorem, which says that an orientable even-dimensional manifold with positive curvature is simply connected. In Chapter 11 we shall deal with some more advanced topics in the theory of manifolds with lower sectional curvature bounds.


Riemannian Manifold Fundamental Group Sectional Curvature Constant Curvature Conjugate Point 
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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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