1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Prove the product of orientable manifolds is again orientable

  1. May 1, 2012 #1
    1. The problem statement, all variables and given/known data

    Let M and N be orientable m- and n-manifolds, respectively. Prove that their product is an orientable (m+n)-manifold.

    2. Relevant equations

    An m-manifold M is orientable iff it has a nowhere vanishing m-form.

    3. The attempt at a solution

    I assume I would take nowhere vanishing m- and n-forms f and g on M and N, respectively, and use them to construct an (m+n)-form h on MxN. However I don't know how this construction would proceed. Any help would be much appreciated.
     
    Last edited: May 1, 2012
  2. jcsd
  3. May 1, 2012 #2

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    I'm assuming that the word orientable in the problem statement is just missing. Given an m-form and an n-form there's only one real way to ever construct an m+n form. It might help to remember/prove that the cotangent bundle of MxN is TM*xTN*
     
  4. May 1, 2012 #3
    Thanks for the response. My first thought is to let

    [tex]\varphi(m,n)(x_1,\cdots,x_{m+n})=f(m,n)(x_1,\cdots,x_m)+g(m,n)(x_{m+1},\cdots,x_{m+n})[/tex]

    so that each φ(m,n) is (m+n)-multilinear, and then apply the alternation mapping A to get the antisymmetric multilinear map h(m,n)=A(φ(m,n)), that is,

    [tex](h)(m,n)(x_1,\cdots,x_{m+n})=\frac{1}{(m+n)!}\sum_{ \sigma\in S_{m+n}}(\text{sgn }\sigma)\varphi(m,n)(x_{ \sigma(1)},\cdots,x_{ \sigma (m+n)})[/tex]

    But there is so much about that map which I wouldn't know how to prove. For instance, is h a diffeomorphism with its image? I know that it maps into the set of (m+n)-multilinear maps from R^{(m+n)(m+n)} into R, but is it really surjective like I need? And is it nowhere vanishing? If I knew in advance that the answers to these questions were all "yes," then I wouldn't mind spending a lot of time trying to prove it. But I don't know any of that, and it's very frustrating.

    As to the cotangent bundle, I'm not sure how that would help. In fact I had to look it up on wikipedia, since I've never encountered it before.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Prove the product of orientable manifolds is again orientable
Loading...