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It is considered a differentiable manifold equipped with a pseudo-Riemannianmetric and an almost contact 3-struc\-ture so that an almost contact metricstructure and two almost contact B-metric structures are generated. There areintroduced the so-called associated Nijenhuis tensors for the studiedstructures. It is given a geometric interpretation of the vanishing of thesetensors as a necessary and sufficient condition for the existence of linearconnections with totally skew-symmetric torsions preserving the structure. Anexample of a 7-dimensional manifold with connections of the considered type isgiven.

Natural Connections with Totally Skew-Symmetric Torsion on Manifolds with Almost Contact 3-Structure and Metrics of Hermitian-Norden Type