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Could anyone suggest a book or a paper containing a good, modern treatment to the Cartan's method of moving frames. Especially, I am interested in its use in studying geometric properties of surfaces in Euclidean geomtry and also, higher dimensional Riemannian geometries.

Thanks

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There is a book called Cartan for beginners which is probably ideal for you :) –  Mariano Suárez-Alvarez Mar 28 '12 at 18:54

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up vote 4 down vote accepted

I believe a go-to book for the modern treatment of Cartan's method of moving frames is known as Olver's blue:

Classical Invariant Theory, (by Peter Olver)
Book preview on Google:
http://books.google.ca/books?hl=en&lr=&id=1GlHYhNRAqEC&oi=fnd&pg=PR10&dq=fels+and+olver+classical+invariant+theory&ots=07SSfI7B3s&sig=YiDGL0ABA_EAttdd7iK9HSgCjD8#v=onepage&q&f=false

At least, an elementary introduction is provided in chapter 8 of this book. This will probably be sufficient for your needs. If necessary, the full details are contained in the original papers by Mark Fels and Peter Olver (they were the two who developed this modern treatment), which are found here:
http://www.math.umn.edu/~olver/mf_/mcI.pdf
http://www.math.umn.edu/~olver/mf_/mcII.pdf

The book mentioned by Mariano is indeed another great source. I believe the focus there is on exterior differential systems, so it's a different read than Olver's book. I like to believe that the title is tongue-in-cheek. ;)

Cartan for Beginners: http://www.ams.org/bookstore-getitem/item=gsm-61

table of contents, preface, and selected pages: http://www.math.tamu.edu/~jml/EDSpublic.pdf

Good luck!

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