Grassmannian bundle

Given a $n$ -manifold $M$ we can consider for every $0 < r < n$ a bundle $π : G (r, T M) \to M$ whose fibre at $x$ is the Grassmannian manifold $G (r, T_{x} M)$

G (r, T M)_{x} := G (r, T_{x} M)

We will denote the points of $G (r, T M)$ as pairs $(p, E)$ with $p \in M$ and $E \in G (r, T_{p} M)$ .

Canonical exterior differential system

The space $G (r, T M)$ carries a canonical linear Pfaffian system: the one whose integral manifolds are exactly the lifts to $G (r, T M)$ of mappings

f : N^{r} \to M

where the lift is defined by $\tilde{f} (x) = (f (x), T_{f (x)} f (N))$ .
It can also be defined by

I_{(p, E)} := π^{*} (E^{⊥})

J_{(p, E)} := π^{*} (T_{p}^{*} M)