Geometry of Linear Differential Systems –towards– Contact Geometry of Second Order

where Jx = Gr(Tx(M), n) is the Grassmann manifold of all n-dimensional subspaces of the tangent space Tx(M) to M at x. Each element u ∈ J(M,n) is a linear subspace of Tx(M) of codimension m, where x = π(u). Hence we have a differential system C of codimension m on J(M,n) by putting: C(u) = π−1 ∗ (u) ⊂ Tu(J(M,n)) π∗ −→ Tx(M). for each u ∈ J(M,n). C is called the Canonical System on J(M,n). We can introduce the inhomogeneous Grassmann coordinate of J(M,n) around uo ∈ J(M,n) as folllows; Take a coordinate system U ′; (x1, · · · , xn, z, · · · , z) of M around xo = π(uo) such that dx1∧· · ·∧dxn |uo 6= 0. Then we have the coordinate system (x1, · · · , xn, z, · · · , z, p1, · · · , pn ) on the neighborhood U = {u ∈ π−1(U ′) | π(u) = x ∈ U ′ and dx1 ∧ · · · ∧ dxn |u 6= 0}; of uo by defining functions p α i (u) on U as follows; dz |u= n ∑