Hi, All: I need some help with some "technology" on differential forms, please: 1)Im trying to understand how the hyperplane field Tx[itex]\Sigma[/itex]< TpM on M=[itex]\Sigma[/itex] x S1 , where [itex]\Sigma[/itex] 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 .