Abstract
We apply Williamson's theorem for the diagonalization of quadratic forms by symplectic matrices to sub-Riemannian (and Riemannian) structures on the Heisenberg groups. A classification of these manifolds, under isometric Lie group automorphisms, is obtained. A (parametrized) list of equivalence class representatives is identified; a geometric characterization of this equivalence relation is provided. A corresponding classification of (drift-free) invariant optimal control problems is exhibited.
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