I'm trying to get a comprehensive understanding of similarities and differences of differential forms and vector fields. As I understand, 1-forms are analogous to vector fields in that for a vector field $X \in \Gamma (TM)$ and a 1-form $\xi \in \Gamma (T^*M)$ defined on an n-dimensional manifold $M$ we have
$X: M \rightarrow TM$
$\xi: M \rightarrow T^{*}M$
where $X$, $\xi$ are sections of $TM$ and $T^{*}M$ (resp.), and $X(p)$, $\xi (p)$ are tangent vectors and cotangent vectors (resp) for any $p \in M$.
With this correspondence of 1-forms and vector fields, I'm wondering if there is an equivalent object from the world of vector fields which corresponds to $k$-forms for $k>1$. For a $k$-form $\omega$ defined over $M$ we have
$\omega: M \rightarrow \Lambda^{k}T^{*}M$.
This $k$-form is created by some operation on 1-forms (wedge product), is there an analogous operation defined on vector fields to create some "$k$-vector field"?
I find that in differential geometry there are so many different objects living in different spaces, I am trying to simplify and draw similarities where possible. This is what motivates the above question.
Best Answer
A section of $\bigwedge^kTM$ is called a polyvector field (or multivector field) of degree $k$. Just as you can take the wedge product of differential forms, you can take the wedge product of polyvector fields. Furthermore, there is also the Schouten-Nijenhuis bracket which extends the Lie bracket of vector fields.