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Abstract
The paper studies fundamental questions of the theory of 2F-planar mappings of pseudo-Riemannian spaces equipped with a certain type of affinor structure. Earlier we proved that a pseudo-Riemannian space with a covariantly constant f-structure is a product of two pseudo-Riemannian spaces, one of which is Kӓhler, and that the class of pseudo-Riemannian spaces with a covariantly constant f-structure is closed under the mappings under consideration.
Also, under the condition of covariant constancy of the affinor in the displayed spaces, non-trivial 2F-planar mappings can be of three types: full and canonical I, II types.
Moreover, depending on the type, the 2F-planar mapping induces respectively a geodesic, holomorphically-projective or affine mapping on the components of the product of the corresponding spaces.
Geometric objects which are invariant under the above mappings are constructed, and classes of spaces admitting $2F$-planar mapping onto a flat space are distinguished. We also found their metrics in special coordinate systems.
Then the question arises of whether other classes of spaces which admit $2F$-planar mapping exist, and how to find them.Using the methods developed in the theory of geodesic mappings, we convert the fundamental equations of the main type $2F$-planar mappings of the pseudo-Riemannian spaces with a covariantly constant $f$-structure to the form that allows us to study them effectively. Using this new form, we, in particular, showed that pseudo-Riemannian space with a covariantly constant f-structure in which concircular or quasiconcircular vector field exists, admits a non-trivial 2F-planar mapping of main type. The fundamental theorems of theory of 2F-planar mappings of pseudo-Riemannian spaces with a covariantly constant f-structure are proved. These theorems supply us with a regular method that enables us to decide effectively whether a space (Vn, gij , Fhi) admits 2F-planar mapping or not, and in the affirmative case, we are able to find all the spaces $(\overline{V}_n, \overline{g}_{ij}, \overline{F}^h_i )$ that can serve as images of Vn under the mappings considered.
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