## Smoothness of subRiemannian isometries.(English)Zbl 1370.53030

Let $$M,N$$ be two bracket-generating sub-Riemannian manifolds, and suppose that $$F : M \to N$$ is a (surjective) isometry of the control (Carnot-Carathéodory) distances. Must $$F$$ be smooth with respect to the differentiable structures of the underlying manifolds of $$M,N$$, and hence be a diffeomorphism? The analogous statement for Riemannian manifolds is well known to be true, but the sub-Riemannian case is more difficult since, unlike the Riemannian distance, the control distance need not be smooth.
In this paper, the authors give a positive answer under the assumption that $$M,N$$ are equiregular. Let $$\Delta^1 = \Delta$$ denote the horizontal distribution of $$M$$, and let $$\Delta^{i+1} = [\Delta, \Delta^{i}]$$. We say $$\Delta$$ is equiregular on an open set $$U$$ if the dimension of $$\Delta^i_p$$ is independent of $$p \in U$$. As an immediate generalization of the result, it follows that an isometry $$F : M \to N$$ is smooth on any open set $$U$$ on which the distribution of $$M$$ is locally equiregular; in particular, there exists such $$U$$ which is dense.
The proof proceeds in two steps which are of independent interest. First, the desired result is shown for general sub-Riemannian manifolds (not necessarily equiregular) under the additional hypothesis that there exist smooth volume forms $$\mathrm{vol}_M, \mathrm{vol}_N$$ on $$M,N$$ such that the isometry $$F$$ pushes forward $$\mathrm{vol}_M$$ to $$\mathrm{vol}_N$$, i.e., $$F_* \mathrm{vol}_M = \mathrm{vol}_N$$. The smoothness of $$F$$ can then be shown by a bootstrap argument based on subelliptic estimates for the sub-Laplacian. Second, it is shown that for equiregular sub-Riemannian manifolds, the canonical Popp volume forms satisfy this hypothesis; this is proved by relating the Popp measure to the spherical Hausdorff measure, where the latter depends only on the metric structure.
As a consequence, taking $$N=M$$, the authors show that if $$M$$ is an equiregular sub-Riemannian manifold, then the isometry group of $$M$$ is a finite-dimensional Lie group; moreover, for each compact subgroup $$K$$, there is a Riemannian extension $$g_K$$ of the sub-Riemannian metric of $$M$$ such that $$K$$ embeds in the isometry group of the Riemannian manifold $$(M, g_K)$$. This holds in particular when $$K$$ is the group of isometries fixing a particular point $$p \in M$$.
Another corollary is that if $$M,N$$ are equiregular and connected and a point $$p \in M$$ is fixed, then any isometry $$F : M \to N$$ is uniquely determined by the value of $$F(p)$$ and the action of the differential $$dF$$ on the horizontal space $$\Delta_p$$ at $$p$$.

### MSC:

 53C17 Sub-Riemannian geometry 35H10 Hypoelliptic equations 35H20 Subelliptic equations 22E25 Nilpotent and solvable Lie groups 28A75 Length, area, volume, other geometric measure theory
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