Kernels, and Representations of Diff. Forms.

[WWGD]

Hi, All:

I need some help with some "technology" on differential forms, please:

1)Im trying to understand how the hyperplane field T_x$\Sigma$<
T_pM on M=$\Sigma$ x S¹ , where $\Sigma$
is a surface, is defined as the kernel of the form dθ (the top form on S¹).

I know that T_(x,y)(MxN)≈T_xM(+)T_yM

But this seems to bring up issues of the dual T_p^*M,

i.e., the cotangent bundle of M .

How do we define a linear map on a sum v_m+v_n , each

a vector on the tangent spaces of M,N at x,y respectively? I think this may have

to see with the tensor product, but I'm not sure.

Thanks .

 

[quasar987]

dθ is defined on the product as the pullback by the natural projection M x S -- > S. It's value on v + w is then dθ(w) (for v in TM, w in TS). So it's kernel is TM.

Does this answer your question?