# Kernels, and Representations of Diff. Forms.

1. Nov 12, 2012

### WWGD

Hi, All:

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

1)Im trying to understand how the hyperplane field Tx$\Sigma$<
TpM 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)≈TxM(+)TyM

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

i.e., the cotangent bundle of M .

How do we define a linear map on a sum vm+vn , 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 .

2. Nov 12, 2012

### 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.