Hi, I have an exercise whose solution seems too simple; please double-check my work:(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# Differential Form on Product Manifold

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