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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.)

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    $\begingroup$ Note that to the best of my knowledge, what are called higher order tangents are duals to $I_x/I_x^k$ (as opposed to $I_x/I_x^2$, which are tangent vectors), where $I_x$ is the ideal of smooth functions vanishing at $x$. Which are decidedly not the same as multivectors, but rather, objects representing higher order ODEs. $\endgroup$ – Bence Racskó Sep 16 '17 at 6:39
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    $\begingroup$ I saw several people call it poly-vector fields. $\endgroup$ – Nicolas Hemelsoet Sep 16 '17 at 9:00
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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).

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