The Tanaka--Webster connection for almost

$\mathcal S$-manifolds and Cartan geometry

Antonio  Lotta and Anna Maria Pastore

Dipartimento  di Matematica Universita di Bari,
Via E. Orabona 4, 70125 Bari, Italy


We prove that a CR-integrable almost $\mathcal S$-manifold
admits a canonical linear connection, which is
a natural generalization of the Tanaka--Webster connection
of a pseudo-hermitian structure on a strongly pseudoconvex
CR manifold of hypersurface type.
Hence a CR-integrable almost $\mathcal S$-structure on a manifold is
canonically interpreted as a reductive Cartan geometry, which
is torsion free if and only if the almost $\mathcal S$-structure
is normal. Contrary to the CR-codimension one case, we exhibit
examples of non normal almost $\mathcal S$-manifolds with
higher CR-codimension, whose Tanaka-Webster curvature vanishes.

AMSclassification. 53C25, 53C15, 53B05, 32V05.

Keywords. Almost $\mathcal S$-structure, Tanaka-Webster connection,
                   Cartan connection, CR manifold.