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abnormal extremals adjoint assume bidegree bounded bracket C-C metric Chow theorem cohomology compact complex structure connectivity theorem contact complex contact manifold contact structure curvature define Definition denote diffeomorphism fact follows formulas G Jk geodesic geometry Hamiltonian Hausdorff Heisenberg group homeomorphism horizontal connection horizontal curves hypoelliptic implies inner product integral curve invariant Jf Jf Jn+1 joining Lemma length Let a G Levi-Civita connection Lie algebra Lie group linear linear span maximal hypoelliptic metric g metric spaces minimizers multi-index neighbourhood normal biextremal null characteristic one-parameter group orbit orthogonal orthonormal framing parametrized by arc-length partial connection piecewise smooth curve projection Proof Proposition prove pseudohermitian connection resp respect Riemannian metric Rumin Complexes satisfies sequence smooth distribution span TM sub-Riemannian manifold sub-Riemannian metric subbundle subset TcCp vector bundle vector fields zero