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Parallelizable Manifold

  1. Oct 31, 2007 #1
    Hi, I am new to manifold and having a hard time on it. :frown: Could anyone please help me on the following problem. Please write down your thoughts. Thanks alot.

    Prove that (S^n) X R is parallelizable for all n.
     
  2. jcsd
  3. Nov 1, 2007 #2
    Show us what you've done so far and tell us precisely where you're stuck.
     
  4. Nov 1, 2007 #3
    Hi,I tried to show S^nXR is parallelizable by showing S^nXR is diffeomorphic to R^n+1\{0} so if R^n+1\{0} is parallelizable then the problem solved. But I just don't know how to show R^n+1\{0} is parallelizable.
     
  5. Nov 6, 2007 #4
    R^(n+1)\{0} is a subset of R^(n+1). You can explicitly write down a global trivialization for R^(n+1) that restricts to a global frame for R^(n+1)\{0}.

    A global frame for R^(n+1) is...

    (1,0,...,0)
    (0,1,0,...,0)
    .
    .
    .
    (0,...,0,1)
     
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