Are there higher-order vector fields (analogously to higher-order differential forms)?

Question

Let $M$ be a smooth manifold.

First,

• a differential $1$-form on $M$ is a smooth section of the cotangent bundle $T^*M$ of $M$
• a vector field on $M$ is a smooth section of the tangent bundle $TM$ of $M$.

Now,

• a differential $k$-form on $M$ is a smooth section of the $k$-th exterior power $\Lambda^k (T^*M)$ of $T^*M$.

So, what do you call a smooth section of the $k$-th exterior power $\Lambda^k(TM)$ of $TM$? Would it be a "higher-order" vector field? Specifically, is it a multivector "field", in the spirit of geometric algebra?

(Please correct any mistakes.)

Answer 1

Sometimes people consider "multi-vector" fields, ie section of $\wedge ^k TM$. Fields of "bi-vectors" are particularly useful in the study of Poisson structures, see https://en.wikipedia.org/wiki/Poisson_manifold.

On these fields, Nijenhuis defined an operation called bracket (https://en.wikipedia.org/wiki/Nijenhuis%E2%80%93Richardson_bracket).