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Geometrical objects associated to a substructure

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 forms, firstly in a global description and secondly using the decomposition in distributions. | Turk J Math 35 (2011) , 717 – 728. ¨ ITAK ˙ c TUB doi:10.3906/mat-0710-33 Geometrical objects associated to a substructure ¨ Fatma Ozdemir, Mircea Crˆ a¸sm˘ areanu Abstract 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 forms, firstly in a global description and secondly using the decomposition in distributions. A generalization of both Hermitian and anti-Hermitian geometry is proposed. Key Words: Polynomial substructure, Induced polynomial structure, Schouten and Vr˘ anceanu connections, (anti)Hermitian metric, Shape operator. 1. Introduction There are plenty of substructures in differential geometry. Usually we are interested in substructures generated by a given structure, e.g. the fruitful theory of submanifolds. In this paper we adopt a different point of view. Namely, we start with a fixed substructure, defined as a tensor field of (1, 1)-type on a given distribution, and search what kind of geometrical objects can be associated in a natural way using a complementary distribution. Based on the fact that the bundle of tensor fields of (1, 1) provides a very interesting geometry (cf. [3], [4], [5]), we obtain a lot of geometric notions, namely: global tensor fields of (1, 1)-type, linear connections, (anti)Hermitian metrics. The paper is divided into three Sections. Firstly, polynomial structures generated by polynomial substructures are studied. The notion of polynomial structure was introduced by Samuel I. Golberg and Kentaro Yano in [10] as a generalization of f -structures (i.e. f 3 +f = 0 ) and studied by various authors