Differential Form on Product Manifold

[WWGD]

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?

 

[Greg Bernhardt]

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?