Several geometric objects, namely global tensor fields of (1, 1) -type, linear connections and Riemannian metrics, associated to a given substructure on a splitting of tangent bundle, are studied. From the point of view of lifting to entire manifold, two types of polynomial substructures are distinguished according to the vanishing of not of the sum of the coefficients. Conditions of parallelism for the extended structure with respect to some remarkable linear connections are given in two

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... firstly in a global description and secondly using the decomposition in distributions. A generalization of both Hermitian and anti-Hermitian geometry is proposed.