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A quasi-linear manifolds and quasi-linear mapping between them

In this article a special class of Banach manifolds (called QL-manifolds) and mapping between them (QL-mappings) are introduced and some examples are given. | Turk J Math 28 (2004) , 205 – 215. ¨ ITAK ˙ c TUB A Quasi–Linear Manifolds and Quasi–Linear Mapping Between Them Akif Abbasov Abstract In this article a special class of Banach manifolds (called QL-manifolds) and mapping between them (QL-mappings) are introduced and some examples are given. 0. Introduction We further develop in this article the theory of QL- mappings, which was started by A. I. Shnirelman ([5]), continued by M.A.Ephendiev ([3]) and also by myself ([1]). As was proved in [1], the classes FQL and FSQL-mappings coincide; however the latter class is more adapted to expansion on affine bundles, which are used in definition of the QL-manifold. As an example, we introduce a QL-manifold structure on the Banach manifold Hs (S 1 , S 2 ). This example shows that QL-manifold structures can be introduced on various classes of mappings. As an example of a QL-mapping, we can take Ff : Hs (S 1 , S 2 ) → Hs (S 1 , S 2 ), where f : S 2 → S 2 is diffeomorphism. We provide definitions of FQL and FSQL-mappings in the appendix. 1. Definitions Let X be a real infinite-dimensional Banach manifold, and {Xj }, Xj−1 ⊂ Xj , j = 1, 2, . is a system of open sets, exhausting X, i.e. X = ∪Xj . Let us suppose ξj = Yj , ψj , Bnj is an affine bundle, where Yj is a total space, Bnj is a basis which is a finitedimensional manifold with boundary, and ψj : Yy → Bnj is the continuous epimorphism. Let Ωj be a bounded domain in Yj , ϕj : Xj → Ωj be a homeomorphism. (ϕj , Xj ) will 205 ABBASOV be called a chart on X. After carrying out the conditions given above we say that on Xj a linear (L−) structure is introduced. If a L−structure is defined on Xj+1 , then obviously, it has been defined on Xj , too (as an induced structure). If ϕj / : Xj / → Ωj / , ϕj // : Xj // → Ωj // , j / , j // ≥ j, are two L−structures on Xj , then the mappings of : Ωj / → Ωj // and ϕj / ◦ ϕ−1 : Ωj // → Ωj / will arise. Let us consider transition ϕj // ◦ ϕ−1 j/ j // them in charts of affine