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Does a compact manifold always have bounded sectional curvature?

  1. Dec 9, 2013 #1
    Sorry if this question seems too trivial for this forum.

    A grad student at my university told me that a compact Riemannian manifold always has lower and upper curvature bounds.

    Is this really true? The problem seems to be that I don't fully understand the curvature tensor's continuity etc.

    What makes me a little skeptical is that I already spent quite a lot of time trying to find a source where this is explicitly stated, without any success. Usually I would expect such a statement as basic as this to appear in lots of books or lecture notes.
     
  2. jcsd
  3. Dec 9, 2013 #2

    lavinia

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    Sectional curvature is defined on tangential 2-planes to the manifold. It is just the Gauss curvature of each plane. So it is defined on a different manifold, the manifold of tangential 2 planes, not the original manifold. Each fiber is itself a compact manifold that is diffeomorphic to what is called the Grassmann manifold of 2 planes and the set off all of them across the entire manifold is itself a compact manifold. So you just need to convince yourself that the sectional curvature of is a continuous function on the manifold of tangential 2-planes since this manifold is compact.
     
  4. Dec 10, 2013 #3
    Thank you!

    Yes, I wasn't sure whether the sectional curvature is a continuous function on the unit tangent bundle.
     
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