Exterior Differential Systems ∗

We will denote by 〈γ1, ..., γk〉 the differential ideal generated by γ1, ..., γk, i.e. the set of elements of Ω∗(M) of the form α ∧ γ1 + ...+ α ∧ γk + β ∧ dγ1 + ...+ β ∧ dγk, for some α, ..., α, β, ..., β ∈ Ω∗(M). We also denote by 〈γ1, ..., γk〉alg the “algebraic” ideal (not necessarily closed under d) consisting of elements of the form α ∧ γ1 + ...+ α ∧ γk. The basic problem of EDS is to find integral submanifolds of M , i.e. submanifolds ι : N ↪→M such that ι∗I = (0). Example 1.2 Consider the system of ordinary differential equations y′(x) = (xyz) z′(x) = cosh(x+ y + z). ∗Notes for a talk given on 2/20/14 at Stanford University. Our primary reference is [Bry]