Riemannian almost CR manifolds with torsion

Abstract

We characterize and study Riemannian almost CR manifolds admitting characteristic connections, that is, metric connections with totally skew-symmetric torsion parallelizing the almost CR structure. Natural constructions are provided of new nontrivial examples. We study the influence of the curvature of the metric on the underlying almost CR structure. A global classification is obtained under flatness assumption of a characteristic connection, provided that the fundamental 2-form of the structure is closed (quasi Sasakian condition).

Generalizations of 3-Sasakian manifolds and skew torsion

“…Notice that for any structure ϕ ∈ Σ M , setting J := ϕ| H : H → H, one has J 2 = −I. Therefore, (H, J) is an almost CR structure on M , compatible with the Riemannian metric g. In [DL14] the authors study metric connections with skew torsion on a Riemannian manifold (M, g) endowed with a compatible almost CR structure (H, J). The characteristic connections defined in that case are required to parallelize the structure (H, J).…”

Some Einstein nilmanifolds with skew torsion arising in CR geometry

“…Theorem 1 [3] Let (M, HM, J, g) be a Riemannian CR manifold. Then M admits a characteristic connection if and only if the following conditions are satisfied:…”

“…Here L denotes the Lie derivative, and σ denotes a cyclic sum. The components of the torsion in ( 3) and ( 4) are determined by the Levi-Tanaka forms L and L of the distributions HM and HM ⊥ (for more details see [3]). We point out that if k < 3 there exists a unique characteristic connection.…”

“…Then one gets q F (z) = k i=1 a i z 2 for every z ∈ C n . Hence it is not hard to see that for every n ≥ 2 and k ∈ {3, 6, 7}, k ≤ n 2 , there exist quadrics of type (n, k), which satisfy (6) Choosing W = {a ∈ H 3 | a =ā} or W = {a ∈ H 3 | a 11 = a 22 = a 33 } we obtain an example of type (3,6) and of type (3,7), respectively.…”

“…In [3], we provided necessary and sufficient conditions for a Riemannian (almost) CR manifold to admit a characteristic connection (Theorem 1); we also furnished an explicit description of the torsion of any characteristic connection. This result generalizes and unifies Theorems 10.1 and 8.2 in [4], concerning almost Hermitian and almost contact metric manifolds.…”