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Keywords:
almost $\mathcal S$-structure; Tanaka–Webster connection; Cartan connection; CR manifold
almost $\mathcal S$-structure; Tanaka–Webster connection; Cartan connection; CR manifold
Summary:
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.
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