Vol. 227, No. 1, 2006

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The tangent grupoid of a Heisenberg manifold

Raphaël Ponge

Vol. 227 (2006), No. 1, 151–175

As a step towards proving an index theorem for hypoelliptic operators on Heisenberg manifolds, including for those on CR and contact manifolds, we construct an analogue for Heisenberg manifolds of Connes’ tangent groupoid of a manifold. As is well known for a Heisenberg manifold (M,H) the relevant notion of tangent bundle is rather that of a Lie group bundle of graded 2-step nilpotent Lie groups GM. We define the tangent groupoid of (M,H) as a differentiable groupoid GHM encoding the smooth deformation of M × M to GM. In particular, this construction makes a crucial use of a refined notion of privileged coordinates and of a tangent-approximation result for Heisenberg diffeomorphisms.

differentiable groupoid, Heisenberg group, foliation, contact structure, CR structure
Mathematical Subject Classification 2000
Primary: 58H05
Secondary: 53C10, 53D10, 32V05
Received: 30 August 2004
Revised: 9 May 2005
Accepted: 12 May 2005
Published: 1 September 2006
Raphaël Ponge
Department of Mathematics
Ohio State University
Columbus, OH 43210-1174
United States