Introduction to exterior differential systems
These lectures explain how to apply the Cartan–Kähler theorem to problems in differential geometry. We want to decide if there are submanifolds of a given dimension inside a given manifold on which given differential forms vanish. The Cartan–Kähler theorem gives a linear algebra test: if the test passes, such submanifolds exist. I will not give a proof or give the most general statement of the theorem, as it is difficult to state precisely.