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.
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$\begingroup$ What do you need to know that is not explained on the Wikipedia page (found easily by search engine, it would seem)? $\endgroup$– Yemon ChoiCommented Jan 18, 2015 at 15:22
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$\begingroup$ I guess I need an introductory reference with a bunch of examples. $\endgroup$– AxiomCommented Jan 18, 2015 at 15:23
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1$\begingroup$ Does this paper satisfy your needs? arxiv.org/abs/1304.7493 $\endgroup$– Joonas IlmavirtaCommented Jan 18, 2015 at 16:08
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$\begingroup$ Yes thanks. That's what I was looking for. $\endgroup$– AxiomCommented Jan 18, 2015 at 16:14
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$\begingroup$ From the algebraic point of view and a characterization among other nilpotent Lie groups, see arxiv.org/abs/1403.5295 (esp. 3.2 for the basic facts). $\endgroup$– YCorCommented Jan 18, 2015 at 22:13
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3 Answers
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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.
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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.
answered Jan 18, 2015 at 16:46
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MR1421823 Gromov, Mikhael Carnot-Carathéodory spaces seen from within. Sub-Riemannian geometry, 79–323, Progr. Math., 144, Birkhäuser, Basel, 1996.
answered Jan 18, 2015 at 16:53
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$\begingroup$ The post above would be interesting for your request: fabricebaudoin.wordpress.com/2013/01/05/… $\endgroup$– ZbigniewCommented Mar 30, 2015 at 15:00
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$\begingroup$ @Zbignev: you entered your comment in the wrong place. You surely wanted to comment on the question, not on my answer. $\endgroup$ Commented Mar 30, 2015 at 18:05
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$\begingroup$ Eremeko I do apologize, I'm not a frequent user of the mathoverflow. $\endgroup$– ZbigniewCommented Apr 1, 2015 at 3:49