I'm trying to learn the Pansu differentiability theorem and I need to know what Carnot groups are. Can someone please explain what Carnot groups are? An introductory reference would be greatly appreciated as well.
Thanks.
I'm trying to learn the Pansu differentiability theorem and I need to know what Carnot groups are. Can someone please explain what Carnot groups are? An introductory reference would be greatly appreciated as well.
Thanks.
I learned the theory of Carnot group or more general subRiemannian manifolds from the thesis of Monti, which can be found here: http://www.math.unipd.it/~monti/PAPERS/TesiFinale.pdf In particular, Section 2 is devoted to the proof of Pansu-differentiability theorem.
Another good notes is by le donne, which can be found here . However the proof of Pansu differentiability theorem is stated without a proof.
By the way, the reference provided above "Gromov, Mikhael Carnot-Carathéodory spaces seen from within. Sub-Riemannian geometry, 79–323, Progr. Math., 144, Birkhäuser, Basel, 1996." is also a nice research suvery.
As an introductory reference, I'm a fan of the book Stratified Lie Groups and Potential Theory for their Sub-Laplacians by A. Bonfiglioli, E. Lanconelli and F. Uguzzoni.
MR1421823 Gromov, Mikhael Carnot-Carathéodory spaces seen from within. Sub-Riemannian geometry, 79–323, Progr. Math., 144, Birkhäuser, Basel, 1996.