Consider smooth vector fields $X_1,..,X_k$ in ${\mathbb R}^n$, satisfying the H\"ormander condition, i.e. for all $x$, the Lie algebra generated by $X_1(x),...,X_k(x)$ is ${\mathbb R}^n$. Do you know if there is any reasonable version of the co-area formula, involving the derivatives with respect to $X_1,...,X_k$ instead of the usual gradient ? Thank you very much! Toccata.
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$\begingroup$ Are you thinking of the coarea formula in terms of the level sets of a single function $u$ on $\mathbb{R}^n$ and wanting to figure out a natural measure multiplier that would only depend on the subRiemannian metric for which the $X_k$ are orthonormal and span a $k$-dimensional distribution that satisfies the Hörmander condition? This can certainly be done because the subRiemannian metric can be used to define a Riemannian metric in a canonical way, and then you can use the ordinary coarea formula for the associated Riemannian metric. But maybe that is too crude for your (unstated) purposes. $\endgroup$– Robert BryantCommented Feb 21, 2013 at 1:41
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$\begingroup$ Thank you very much for your answer! Yes, this is what I am thinking about. Do you know any reference for that ? Thanks again. Toccata. $\endgroup$– ToccataCommented Feb 21, 2013 at 22:12
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$\begingroup$ @Toccata: I don't know references that specifically discuss the coarea formula in the subRiemannian case, but the fact that one can associate a Riemannian metric in a natural way to a subRiemannian metric satisfying Hörmander's condition and some extra regularity is well-known and seems to have been re-discovered independently by a number of different people. I think you can find a general discussion of it in some of U. Hammenstadt's papers. For a simple, specific example of such a construction, see Hughen's 1995 thesis, available at montgomery.math.ucsc.edu/papers/HughenThesis.pdf $\endgroup$– Robert BryantCommented Feb 22, 2013 at 14:49
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Yes, a generalization of Federer’s co-area formula is known for the horizontal gradient in sub-Riemannian geometry.
This is proved under mild conditions for instance in the paper
N. Garofalo, D.M. Nhieu, Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot-Caratheodory spaces, J. Anal. Math., 74, (1998), 67-97.
You may also have a look in these Lecture notes by P. Pansu, Section 2.6,, where the co-area formula is used to prove the equivalence between the Sobolev inequality and the isoperimetric inequality in sub-Riemannian manifolds.
Submanifolds and differential forms on Carnot manifolds, after M. Gromov and M. Rumin
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1$\begingroup$ Pansu's notes is too sketch and no details are given. But of course, he always has the correct intuition. $\endgroup$ Commented Feb 26, 2015 at 8:17