R W Sharpe, Differential Geometry: Cartan's Generalization of Klein's Erlangen Program of Klein's Erlangen Program: A Review: A Review - Post 2

After a first reading of Sharpe's book, I wanted to apply the wealth of new results to General Relativity and Gauge theories. However, I felt that I needed to assimilate the material further. So I decided to let it incubate in the subconscious till the time it took for the new to get integrated with the already established patterns within. Around this time, I discovered Sardanashvily's beautiful papers and three books 'New Lagrangian and Hamiltonian Methods in Field Theory', 'Gauge Mechanics' and 'Connections in Classical and Quantum Field Theory.' Due to the computational nature of these books, I soon mastered the Jet manifold methods, especially from the last of these which I worked out in full. 

This gave me a perspicacious understanding of Cartan's method of prolongation. One could set forth a vector field or differential form as sections of vector or and its dual bundles and then prolong it to higher orders by lifting it up. In the naive terms, one takes a vector field on the base space and adds another 'vertical' vector field, the connection to it and forms the lifted vector field. If one takes the 'cross-product' or bracket on the base space alone, the result is trivial. The cross-product or the bracket vanishes. The 'cross-product' or bracket of lifted vector field on the full space, taken with respect to the full space, however, need not, in general , vanish and the result is the curvature or 'field strength' or the 'correction' term or what have you depending on how one interprets the base vector field. One may prolong the base quantities to other orders and obtain elegant formulations of several differential oerators like the Euler-Lagrange operator. The vanishing of the Euler-Lagrange equations for physical motions can now be recast as the vanishing of the Euler-Lagrange operator when its lift is contracted with the interior operator i.e when it is pulled-back to the actual path. 

Robert Hermann also has a fine treatment of this. Other highly instructive treatments are to be found in Goldschmidt and Sternberg, 'The Hamilton-Cartan formalism in the calculus of variations, Annales de l'institut Fourier, 23 no. 1 (1973), p. 203-267 and the GIMMSY papers, 'Momentum Maps and Classical Relativistic Fields', Binz, Sniyatzki and Fischer, 'Geometry of classical fields', a series of articles by A. Echeverría-Enríquez et al and by Aladya and Azcarraga. In all these fine sources, what is missing is the original, raw, intuitive meaning and geometrical motivation so much necessary for the learning process. Sharpe's book has this in abundance. From there, I could trace out further lines of thought in Chern's writings. As always, Chern's writings illuminate every step of the path and the preface he has written to Sharpe's book does just that.

The central theme of Sharpe's book is Cartan's insight that Euclidean geometry could be generalized in two different ways. The first by replacing it by a study of the euclidean group of rigid motions. The second by localizing it. Now the first may be followed by the second or the second by the first. Whatever be the order, one obtains what we may call a Localized-Group Geometry now termed a Cartan Geometry. This observation is profound in its implications. Sharpe goes on to treat different formulations and variants of this scheme.

As soon as I understood this, I was struck by the scope of this generalization. I realized that I could take anything, whatsoever, as a starting object and apply this scheme to it. I could take Minkowski space and arrive at a Metric Affine Gauge theory via the Poincare group followed by its localization. I could start with Newtonian mechanics and obtain a gauge mechanics via the Galilean group followed by its localization. I could take a space and obtain a non-trivial group manifold via a group followed by its localization and so on. A similar thing was possible by going over to the Hamiltonian formalism and Symplectic geometry. 

I began to assemble diverse results that I had began to get as upsurgings from the subconscious. The incubation was complete and it was time to verify the results. I began to write them down and was just getting into the flow when I had to interrupt it due to formal academic requirements that made considerable demands on my time and attention. Most reluctantly I set aside my writing and focused on the academics. When I finally managed to return to it, I found to my utter dismay that though memories remained, the dynamism of the insights had diminished and the flow nowhere in sight. 

Reading Sharpe's book for the fourth time, suddenly connected back to the insights. I realized due to my present involvement in Cognitive Neuroscience, that such a thing is not accidental. It is possible to recover past experiences in full by triggering other associated experiences. Therefore, as soon as I stumbled upon few of the insights, I put myself into a partial trance on the associated experience of reading Sharpe's book. In addition, writing is a powerful means to channelize the subconscious stirrings into conscious form. This is my main reason for writing in this blog.