Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems

This graduate level text is a second edition of the authors’ very successful first edition published in 2003. There are some significant differences from the first edition including three chapters of essentially new material (chapter 3 on Riemannian geometry, chapter 10 on Darboux integrable exterior differential systems and chapter 11 on conformal differential geometry). The book unites differential geometry and partial differential equations via Elie Cartan’s method of moving frames and the theory of exterior differential systems (EDS). The text may be used as a semester or year-long course with different possible emphases including differential geometry, EDS with PDE applications, moving frames in algebraic geometry and differentiable manifolds.

An exterior differential system is, roughly speaking, an expression of a differential equation in the language of differential forms. For example, \[ \frac{dy}{dx}=f(x) \] expressed as \(dy-f(x)dx=0\). “Solving” the differential equation becomes equivalent to finding “where” the differential form vanishes. Using the language of jet bundles, EDS permits formulations of PDEs without coordinates.

Elie Cartan pioneered the method of moving frames as a coordinate free way of studying differential geometry. A moving frame is a basis of vectors (tangent, movement, directional etc.) at each point of a curve, surface, or manifold. If the manifold is Riemannian (has a Riemannian metric), one considers orthonormal bases. One of the simplest examples is the familiar unit tangent, normal and binormal vectors \(\mathbf{T}\), \(\mathbf{N}\), \(\mathbf{B}\) from elementary calculus. (The Frenet-Serret frame.) In elementary calculus students learn that differential invariants such as curvature and torsion are revealed by differentiating these vectors with respect to arc length. The text by Ivey and Landsberg is of course considerably more advanced. EDS also lends itself to differential geometry, a classical example being the Cartan-Janet theorem (proved with EDS) which deals with local isometric embedding of analytic Riemannian manifolds. Moreover, many PDEs carry important differential geometric structure which can be revealed through EDS computations. The entire chapter 7 is devoted to applications to PDE.

It should be pointed out that this book is rather advanced and presumes some prior familiarity with classical differential geometry and differentiable manifolds. There are a few companion resources a newcomer to the field might find helpful. Robert Bryant’s unpublished class lecture notes, untitled but locatable through a Google search, present a somewhat gentler approach to EDS. A recent book by Jeanne N. Clelland, From Frenet to Cartan: The Method of Moving Frames presents a somewhat gentler approach to the differential geometry. In effect, Ivey and Landsberg combine these two topics. A somewhat more advanced, but very carefully written supplement would be the Springer publication Exterior Differential Systems by Bryant, Chern, Gardner, Goldschmidt, Griffiths availabe for free download at MSRI.

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Steven Deckelman is a professor of mathematics at the University of Wisconsin-Stout, where he has been since 1997. He received his Ph.D from the University of Wisconsin-Madison in 1994 for a thesis in several complex variables written under Patrick Ahern. Some of his interests include complex analysis, mathematical biology and the history of mathematics.