Let $V$ be a vector space, then denote the simple or decomposable $m$-vectors in $\bigwedge^m V$ by $GC_m(V)$.

I am struggling to understand the topology of the bundle $GC_m(TN) = \bigsqcup_{q\in N} GC_m(T_qN)$ of a smooth $n$-dimensional manifold $N$. What kind of bundle is this even?

I know that there is the family of Plücker embeddings $\rho_q: G_m(T_qN) \to \mathbb{P}(\bigwedge^mV)$ from the Grassmannian manifold over the tangent spaces to the exterior power. Its image is the projective image of the simple $m$-tangent vectors. So my initial thoughts were to first understand the topology of the bundle $G_m(TN) = \bigsqcup_{q\in N} G_m(T_qN)$ and then just transferring this topology via $\rho_q$ to $GC_m(TN)$.

As this is the first time fibre bundles cross my path I find it hard to even show that $G_m(TN)$ is a fibre bundle. Are there any references where this is explicitly shown or is this trivial? How would I define the transition functions for this bundle? What's the group action on the manifold $G_m(T_qN)$?

PS: My questions originates from Finsler geometry where $m$-volume densities of a Finsler manifold $N$ are defined as continuous functions from $GC_m(TN)$ to $\mathbb{R}$, hence my question about the topology.


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