Nine Lectures on Exterior Diff erential Systems Informal notes for a series of lectures delivered 1223 July 1999 at the Graduate Sum- mer Workshop on Exterior Diff erential Systems at the Mathematical Sciences Research Institute in Berkeley, CA. by Robert L. Bryant Duke University Durham, NC bryantmath.duke.edu 1 Lecture 1. Basic Systems 1.1. What is an exterior differential system? An exterior diff erential system (EDS) is a pair (M,I) where M is a smooth manifold and I (M) is a graded ideal in the ring (M) of diff erential forms on M that is closed under exterior diff erentiation, i.e., for any in I, its exterior derivative d also lies in I. The main interest in an EDS (M,I) centers around the problem of describing the submanifolds f : N M for which all the elements of I vanish when pulled back to N, i.e., for which f = 0 for all I. Such submanifolds are said to be integral manifolds of I. (The choice of the adjective integral will be explained shortly.) In practice, most EDS are constructed so that their integral manifolds will be the solutions of some geometric problem one wants to study. Then the techniques to be described in these lectures can be brought to bear. The most common way of specifying an EDS (M,I) is to give a list of generators of I. For 1,.,s (M), the algebraic ideal consisting of elements of the form = 11+ ss will be denoted ?1,.,s?alg while the diff erential ideal I consisting of elements of the form = 11+ ss+ 1d1+ sds will be denoted ?1,.,s?. Exercise 1.1: Show that I = ?1,.,s ? really is a diff erentially closed ideal in (M). Show also that a submanifold f : N M is an integral manifold of I if and only if f= 0 for = 1,.,s. The p-th graded piece of I, i.e.,I p(M), will be denoted Ip.For any x M, the evaluation of p(M) at x will be denoted xand is an element of p x(M) = p(T xM). The symbols Ix and Ip x will be used for the corresponding concepts. Exercise 1.2:Make a list of the possible ideals in (V ) up to isomorphism, where V is a vector space over R of dimension at most 4. (Keep this list handy. Well come back to it.) 1.2. Differential equations reformulated as EDSs Elie Cartan developed the theory of exterior diff erential systems as a coordinate-free way to describe and study partial diff erential equations. Before I describe the general relationship, lets consider some examples: Example 1.1: An Ordinary Diff erential Equation. Consider the system of ordinary diff erential equations y?= F(x,y,z) z?= G(x,y,z) where F and G are smooth functions on some domain M R3. This can be modeled by the EDS (M,I) where I = ?dy F(x,y,z)dx, dz G(x,y,z)dx?. Its clear that the 1-dimensional integral manifolds of I are just the integral curves of the vector fi eld X = x + F(x,y,z) y + G(x,y,z) z . 2 Example 1.2: A Pair of Partial Diff erential Equations. Consider the system of partial diff erential equations zx= F(x,y,z) zy= G(x,y,z) where F and G are smooth functions on some domain M R3. This can be modeled by the EDS (M,I) where I = ?dz F(x,y,z)dx G(x,y,z)dy?. On any 2-dimensional integral manifold N2 M of I, the diff erentials dx and dy must be linearly indepen- dent (Why?). Thus, N can be locally represented as a graph ?x,y,u(x,y)? The 1-form dzF(x,y,z)dxG(x,y,z)dy vanishes when pulled back to such a graph if and only if the function u satisfi es the diff erential equations ux(x,y) = F ?x,y,u(x,y)? uy(x,y) = G?x,y,u(x,y)? for all (x,y) in the domain of u. Exercise 1.3:Check that a surface N M is an integral manifold of I if and only if each of the vector fi elds X = x + F(x,y,z) z andY = y + G(x,y,z) z is tangent to N at every point of N. In other words, N must be a union of integral curves of X and also a union of integral curves of Y . By considering the special case F = y and G = x, show that there need not be any 2-dimensional integral manifolds of I at all. Example 1.3:Complex Curves in C2.Consider M = C2, with coordinates z = x + iy and w = u + iv. Let I = ?1,2? where 1and 2are the real and imaginary parts, respectively, of dzdw = dxdu dydv + i(dxdv + dydu). Since I1= (0), any (real) curve in C2is an integral curve of I. A (real) surface N C2is an integral manifold of I if and only if it is a complex curve. If dx and dy are linearly independent on N, then locally N can be written as a graph ?x,y,u(x,y),v(x,y)? where u and v satisfy the Cauchy-Riemann equations: ux vy= uy+ vx= 0. Thus, (M,I) provides a model for the Cauchy-Riemann equations. In fact, any reasonable system of partial diff erential equations can be described by an exterior diff er- ential system. For concreteness, lets just stick with the fi rst order case. Suppose, for example, that you have a system of equations of the form F?x,z, z x ? = 0, = 1,.,r, where x = (x1,.,xn) are the independent variables, z = (z1,.,zs) are the dependent variables, and z x is the Jacobian ma