MATH 848I: Exterior Differential Systems

Spring 2020

This course will be an introduction to Cartan's method of moving frames and EDS. The latter is a method for encoding systems of partial differential equations on manifolds as differential ideals on associated manifolds, which is particularly adapted to the intrinsic geometry of the PDEs. The core of the course will be based on Ivey and Landsberg's Cartan for Beginners Chapters 1 and 4-7 (first edition)/Chapters 1 and 5-8 (second edition), which includes the Cartan-Kähler algorithm yielding a criterion for existence of solutions with given initial conditions, as well as applications to particular classes of PDEs such as Monge-Ampère systems.

Suggested Texts:

T.A. Ivey and J.M. Landsberg: Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems (2nd ed.), AMS Graduate Studies in Mathematics 175, Providence, RI (2016).
R. Bryant, P. Griffiths, D. Grossman: Exterior Differential Systems and Euler-Lagrange Partial Differential Equations, Chicago Lectures in Mathematics, Chicago (2003).
R. Bryant, S.-S. Chern, R.B. Gardiner, H. Goldschmidt, P. Griffiths: Exterior Differential Systems, Springer (1990).

Announcements

Class meets Tu-Th 11-12:15 in Math 1311.
In preparation for possible online classes, please make sure you can access the course canvas page, and make contact with me on Skype.
Lectures will be online starting the week of March 30. The current plan is to post prerecorded lectures on the class canvas page and to hold discussion sessions during class time on Thursdays. My primary mode of communicating plans to the class will be through email.

Lectures

1/28: Examples of overdetermined PDEs (1.3 in IL, see also App. B1, B2 for a review of differential forms)
1/30: Differential ideals and the Frobenius Theorem (1.10 in IL)
2/4: Maurer-Cartan form (1.6 in IL, see also App. A.4 for Lie algebras)
2/6: Fundamental Theorem for the Maurer-Cartan form (1.6 in IL; see handout for proof of last point of theorem)
2/11: Examples of moving frames: curves in the Euclidean plane, maps between complex domains (1.4, 1.7 in IL)
2/13: Tableaux and prolongations (5.1 in IL)
2/18: Solutions of tableaux in the form of the "first example" (5.2 in IL)
2/20: Solutions of tableaux in the form of the "second example" (5.3 in IL)
2/25: Class CANCELLED
2/27: Characters of a tableau (5.5 in IL)
3/3: Involutivity and Cartan-Kähler Theorem for Tableaux (5.5, 6.3 in IL)
3/5: Cauchy-Kovalevskaya Theorem (D.2 in IL), integral elements and polar spaces (8.1 in IL)
3/10: Example of non-regular flag for a tableau (5.6.1 in IL); definition of linear Pfaffian system (6.1 in IL); example of tableau (6.3 in IL)
3/12: Example with nonzero torsion and trivial tableau (6.2 in IL); interpretation of intrinsic torsion as complete obstruction to nontriviality of the space of n-dimensional integral elements (where n is the order of J\I); Cartan-Kähler statement for linear Pfaffian systems (theorem 6.5.5 in IL); Linear Pfaffian system for conformal maps between Riemannian surfaces, existence of isothermal coordinates (example 6.8.4 in IL).
3/30: Review of basics of linear Pfaffian systems (IL 6.1); proof of proposition on interpretation of A_x⁽¹⁾ (IL 6.7).
4/1: Framings of orthonormal frame bundles of space forms from Maurer-Cartan forms.
4/2: Canonical framing of the orthonormal frame bundle of a Riemannian manifold (IL 3.1); (References for principal connections: Kobayashi-Nomizu's Foundations of Differential Geometry vol. 1 Ch II (also Chs III, IV); Poor's Differential Geometric Structures; or Sharpe's Differential Geometry App. on Ehresmann connections).
4/3: Curvature 2-form (IL 3.1).
4/6: EDS for isometric immersion of surface in E³, vanishing of torsion, failure of involutivity (IL 6.4)
4/8: Prolongation of EDS for isometric immersion of surface in E³ (IL 6.4)
4/9: Gauss Equation arising from intrinsic torsion of prolonged EDS, calculation of π^εs in restriction to locus of Gauss Equations (Il 6.4)
4/11: Elimination of torsion, verification of involutivity, solution for local isometric embedding of surfaces into E³ (IL 6.4)
4/13: Cartan-Kähler Algorithm for linear Pfaffian systems (IL 6.5, 6.6)
4/14: EDS for isometric immersion with arbitrary dimension and codimension, vanishing of torsion, failure of involutivity, prolongation, Gauss Equations (IL 6.4)
4/15: Cartan-Janet Theorem statement (6.10.1 in IL); linear algebra of Gauss Equations: Aside 6.4.5 in IL, statement of Lemma 6.10.2
4/21: Proof of lemma 6.10.2; introduction to 2nd Bianchi identity for curvature 2-form.
4/23: more on 2nd Bianchi identity for curvature 2-form and relation to ∇ R.
4/27: Proof of Cartan-Janet Theorem (6.10.1 in IL): eliminating torsion.
4/28: Proof of Cartan-Janet Theorem (6.10.1 in IL): estimating characters, involutivity of tableau.
4/29: Cauchy characteristic vector fields (IL 7.1)
5/1: Contact system encoding first-order PDE (IL 1.10); method of Cauchy characteristics to solve first-order PDEs (IL 7.1)
5/3: Inviscid Burgers Equation (IL Example 7.1.13)
5/6: Linear Pfaffian system encoding second-order PDE (IL Example 6.5.3) and corresponding characteristic variety (IL 5.6, 6.7)
5/7: Characteristic systems and Monge characteristics arising from the characteristic variety for a hyperbolic second-order PDE (IL 7.2); Monge Ampère PDEs (IL 7.4)
5/8: Monge-Ampère systems (IL Definition 7.4.3, Exercise 7.4.4, Proposition 7.4.7, Examples 7.4.5, 7.4.6, 7.4.9)
5/12: Linear Weingarten surfaces and Bonnet Theorem (IL 7.4, Exercise 7.4.15); Pseudospherical surfaces (IL Example 7.4.17)
5/13: Psuedospherical surfaces cont'd, and the sine-Gordon Equation (IL Example 7.4.17)