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Differential Form on Product Manifold

  1. Jun 15, 2014 #1

    WWGD

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    Hi, I have an exercise whose solution seems too simple; please double-check my work:

    We have a product manifold MxN, and want to show that if w is a k-form in M and

    w' is a k-form in N, then ##(w \bigoplus w')(X,Y)## , for vector fields X,Y in M,N respectively,

    is a k-form in MxN.

    I am assuming k=1 , and then we can generalize. My proof:

    We start with (the isomorphism):

    ##(T_{(m,n)}(M\times N) = (T_m M \bigoplus T_n N)##,

    Then, dualizing both sides:


    ##(T_{(m,n)} (M \times N))^* =(T_m M \bigoplus T_n N)^*##.

    We then use that ##(A \bigoplus B)^*= A^* \bigoplus B^*## , to get :

    ##(T_{(m,n)} (M \times N))^* = (T_m M )^*\bigoplus (T_n N)^* ##.

    Is that all there is to it?
     
    Last edited: Jun 15, 2014
  2. jcsd
  3. Jul 1, 2014 #2
    I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?
     
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