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Moving Frames and Moving Coframes

First introduced by Gaston Darboux, and then brought to maturity by Élie Cartan, [4], [5], the theory of moving frames (“repères mobiles”) is widely acknowledged to be a powerful tool for studying the geometric properties of submanifolds under the action of a transformation group. While the basic ideas of moving frames for classical group actions are now ubiquitous in differential geometry, the theory and practice of the moving frame method for more general transformation group actions has remained relatively undeveloped. The famous critical assessment by Weyl in his review, [27], of Cartan’s seminal book, [5], retains its perspicuity to this day: “I did not quite understand how he [Cartan] does this in general, though in the examples he gives the procedure is clear. . . . Nevertheless, I must admit I found the book, like most of Cartan’s papers, hard reading.” Implementations of the method of moving frames for certain groups having direct geometrical significance — including the Euclidean, affine, and projective groups — can be found in both Cartan’s original treatise, [5], as well as many standard texts in differential geometry, e.g., [13], [24], [28]. The method continues to attract the attention of modern day researchers and has been successfully extended to a few additional examples, including, for instance, holomorphic curves in projective spaces and Grassmannians. The papers of Griffiths, [12], Green, [11], Chern, [7], and the lecture notes of Jensen, [14], are particularly noteworthy attempts to place Cartan’s intuitive constructions on a firm theoretical and differential geometric foundation.