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Sub-Laplacian-Commuting Maps on Sub-Riemannian Lie Groups

A characterization of the smooth maps between sub-Riemannian Lie groups that commute with the intrinsic sub-Laplacian: such maps are exactly sub-Riemannian conformal submersions, extending the classical Riemannian isometry / harmonic-morphism results to the sub-elliptic setting. A consequence is that on a Carnot group the sub-Laplacian determines the sub-Riemannian structure, illustrated explicitly on Heisenberg groups.