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\(4\)-planar mappings of almost quaternionic and almost antiquaternionic Spaces. (English) Zbl 0969.53006

\(4\)-quasiplanar mappings of almost quaternionic spaces have been studied by I. N. Kurbatova [Sov. Math. 30, 100-104 (1986; Zbl 0602.53029)] and J. Mikeš, J. Némčiková and O. Pokorná [Suppl. Rend. Circ. Mat. Palermo, II. Ser. 54, 75-81 (1997; Zbl 0957.53007)]. These mappings generalize the geodesics, quasigeodesics and holomorphically projective mappings of Riemannian and Kählerian spaces D. V. Beklemishev [Geometria, Itogi Nauki i Techn., All-Union Inst. for Sci. and Techn. Information (VINITI), Akad. Nauk SSSR, Moscow, 165-212 (1965)]; J. Mikeš [J. Math. Sci., New York 78, No. 3, 311-333 (1996; Zbl 0866.53028)]; J. Mikeš [Itogi Nauki i Techniki, Ser. Sovr. Mat. i jego priloenija, T. 30, Geom. 3, VINITI, 240-281 (1995)]; J. Mikeš and N. S. Sinyukov [Sov. Math. 27, 63-70 (1983; Zbl 0526.53013)]; A. Z. Petrov [Grav. i teor. Otnos. Issues 4-5, Kazan State Univ. Press, Kazan, 7-21 (1968)]; N. S. Sinyukov [Nauka, Moscow (1979; Zbl 0637.53020)]and [Itogi Nauki i Techn., Geometrija, Moskva, VINITI 13, 3-26 (1982; Zbl 0498.53010)]. Similar problems have been studied in the context of complex manifolds, see T. N. Bailey and M. G. Eastwood [Forum Math. 3, 61-103 (1991; Zbl 0728.53005)]. In the paper under review, several facts concerning \(4\)-planar mappings of almost quaternionic and almost antiquaternionic spaces are presented in a unified manner. In particular, the fundamental equations of these mappings are established.

MSC:

53B10 Projective connections
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
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