I am seeking a bit of clarification on vector valued forms.

Intuitively, a vector valued differential form is a differential form on $M$ whose values lie in some vector space. More accurately,

Let $\pi:E\rightarrow M$ be a vector bundle. An $E$-valued differential $k$-form $\omega$ is a member of the space

$ \begin{align*}\Gamma\left(E\otimes \Lambda^k(T^*M)\right)&=\Gamma\left(E\otimes \bigsqcup_{q\in M}\Lambda^k(T_q^*M)\right). \end{align*}$

My first question:

Is this to mean that for $q\in M$,

$ \begin{align*} \omega(q)\in \pi^{-1}(q)\otimes(\Lambda^k(T_q^*M))=E_q\otimes (\Lambda^k(T_q^*M))? \end{align*} $

Furthermore, is it true to say that if $(e_i)_{i=1}^n$ is a basis for $E_q$ and if $M$ is $m$-dimensional, then

$ \begin{align*} E_q\otimes (\Lambda^k(T_q^*M))&=\text{span}\left\{e_i\otimes dx^{i_1}\wedge \cdots \wedge dx^{i_k}\right\}, \quad 1\leq i \leq n,\quad 1\leq i_1<\cdots <i_k \leq m, \end{align*} $

Where $e_i\otimes dx^{i_1}\wedge \cdots \wedge dx^{i_k}$ is the multi-linear map

$ \begin{align*} E_q^*\times \underset{k\text{ times}}{\underbrace{{T_q M\times\cdots\times T_q M}}}&\rightarrow \mathbb{R}\\ \\ (\eta^*,X_1,\dots,X_k)&\mapsto e_i(\eta^*)\cdot dx^{i_1}\wedge\dots\wedge dx^{i_k}(X_1,\dots,X_k),\quad \eta^*\in E_q^*,\quad X_1,\dots,X_k\in T_p M. \end{align*} $

I'm not sure if the above is correct - it's the only way I could really make sense of the tensor product of the two spaces. If someone could clarify any misunderstandings that would be great.

My second question:

Is the reason why $\omega$ is a 'vector valued form' because it is a linear combination of the quantities $e_i\otimes dx^{i_1}\wedge \cdots \wedge dx^{i_k}$, thus, we can write $\omega(\cdot,X_1,\dots,X_k)=\alpha^i e_i$. I kind of see this as, the $dx^i$-s spit out a real number after putting the $k$ vectors in and these are just the coefficients of the remaining $e_i$-s.

My overriding question comes from the fact that I have learnt that $V\otimes W$ is the set of linear combinations of $v_i\otimes w_j$ where $(v_i)$ and $(w_j)$ are bases for the vector spaces. So it is logical to me that the tensor product $E_q\otimes (\Lambda^k(T^*_q M))$ is the span of elements of the form `basis of $E_q\,\otimes$ basis of $\Lambda^k(T_q^* M)$' - if that makes any sense at all. Given that the basis of $\Lambda^k(T^*_q M)$ is the set of all $dx^{i_1}\wedge\dots\wedge dx^{i_k}$ such that $i_j$-s are increasing, this is where my question originates.

  • 1
    $\begingroup$ Yes, this is all correct. $\endgroup$ – Jesse Madnick Mar 5 '15 at 5:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.