For a vector field $X$ on a manifold there are two ways to define a Lie derivative: an algebraic one using Cartan's formula $\mathcal{L}_X \alpha = i_X d \alpha + d i_X \alpha$ and a dynamical one using the flow $\phi^X_t$, $${\cal L}_X \alpha = \frac{d}{dt}\biggl|_{t=0} \left(\bigl(\Phi^t_X\bigr)^* T\right).$$
Given a multivector field $X$ one can define its Lie derivative by means of Cartan's formula, i.e, $\mathcal{L}_X \alpha = i_X d \alpha + d i_X \alpha$. See, for example [1].
My broad question is if the Lie derivative by a multivector means something dynamical. These are some fuzzy questions for which any help or reference would be welcomed.
In the simplest case, if $X = X_1\wedge \dotsb \wedge X_n$ and the $X_i$ span an integrable distribution, is $\alpha$ constant in some sense on the leaves of this distribution? What happens if the multivector is not integrable?
Is there a generalization of the concept of flow for a multivector that applies to this situation?
My current geometric understanding of general multivectors is that they are linear combinations of hyperplanes modulo the Plücker relations (whose geometric interpretation feels somewhat obscure to me). I would like to have a better interpretation. On question [2] on this site it is stated that they are global sections of a line bundle over the Grasmannian, some reference of this fact would be useful.
EDIT:
- On [2] it is said that "is known that $Λk(V)$ is isomorphic to the space of global sections of a line bundle over the Grassmanian $Gk(V)$ of k-dimensional subspaces in $V$ - namely of the highest exterior power of the tautological bundle." I am trying to make sense of this. Let $V_k(V^*)$ be the bundle over $G_k(V^*)$ whose elements $(W,a)$ are $k$-dimensional subespaces on $V^*$ and $a$ is a volume form on $W$. A $k$-vector field $X$ can be though of as a section of this bundle: to every subspace we assign the restriction of $X$. If I understand the quotation above and it holds over the reals, $k$-vector fields are presicely algebraic sections of that bundle. Hence, a multivector is an algebraic assignment of a volume to each subespace of $V^*$. Is this true? Any reference would be great.
Also, as a follow-up partially unrelated question. Forms should be then algebraic assignments of volume to subspaces. But why would a differential geometer limit themself to algebraic sections? Has someone studied the sections of this bundle. It is posible to define integrals of submanifolds over such sections?
[2] მამუკა ჯიბლაძე, Cannot multivectors be classified more easily than general tensors?, URL (version: 2017-04-13): Cannot multivectors be classified more easily than general tensors?
