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**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 \(p\).

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 \(p\).

Reviewer: Nathaniel Eldredge (Greeley)

### 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 |