Abstract
We study connections between the Lie bracket on the tangent space of a homogeneous periodic Φ-space and the operators of canonical affinor structures of this space. The relations obtained allowed us to single out several cases of integrability of the structures under consideration.
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References
N. A. Stepanov, “Basic Facts of the Theory of φ-Spaces,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 3, 88–95 (1967).
O. Kowalski, Generalized Symmetric Spaces (Springer, Berlin, 1980; Mir, Moscow, 1984).
N. A. Stepanov, “Homogeneous 3-Cyclic Spaces,” Izv. Vyssh. Uchebn. Zaved.Mat, No. 12, 65–74 (1967).
J. A. Wolf and A. Gray, “Homogeneous Spaces Defined by Lie Group Automorphisms,” J. Diff. Geom. 2(1–2), 77–159 (1968).
A. Gray, “Riemannian Manifolds with Geodesic Symmetries of Order 3,” J.Diff. Geom. 7(3–4), 343–369 (1972).
N. A. Stepanov, “Almost Complex Structures on φ-Spaces,” in Abstracts of the 3rd Sci. Conf. on Problems of Geometry (Kazan, 1967), pp. 158–160.
Gr. Tsagas and Ph. Xenos, “Relation Between Almost Complex Structures and Lie Bracket for a Special Homogeneous Spaces,” Tensor 41(3), 278–284 (1984).
Ph. Xenos, “Properties of the Homogeneous Spaces of Order Five,” Bull. of the Calcutta Math. Soc. 78(5), 293–302 (1986).
V. V. Balashchenko and Yu. D. Churbanov, “Invariant Structures on Homogeneous Φ-Spaces of Order 5,” Usp. Mat. Nauk 45(1), 169–170 (1990).
Yu. D. Churbanov, “Geometry of Homogeneous Φ-Spaces of Order 5,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 5, 70–81 (2002) [Russian Mathematics (Iz. VUZ) 46 (5), 68–79 (2002)].
A. A. Ermolitskii, “Periodic Affinors and 2k-symmetric Spaces,” Dokl. Belarus Akad. Nauk 34(2), 109–111 (1990).
V. V. Balashchenko and N. A. Stepanov, “Canonical Affinor Structures of Classical Type on Regular Φ-Spaces,” Matem Sborn. 186(11), 3–34 (1995).
Yu. D. Churbanov, “The Geometry of Special Affinor Structures of Homogeneous Φ-Spaces of Odd Orders,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 2, 84–86 (1994) [Russian Mathematics (Iz. VUZ) 40, (2), 82–84 (1994)].
Yu. D. Churbanov, “Classical Affinor Structures of Homogeneous Φ-Spaces of Odd Order,” in Abstracts of the VII Belarus Math. Conf. Part 1 (Minsk, 1996), pp. 147–148.
Yu. D. Churbanov, “Affinor Structures of Classical Type of Homogeneous Periodic Φ-Spaces,” in Abstracts of the VIII Belarus Math. Conf. Part 2 (Minsk, 2000), p. 131.
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry (Interscience Publishers, NY, 1963; Nauka, Moscow, 1981), Vol. 1.
O. V. Dashevich, “Canonical Structures of Classical Type on Regular Φ-Spaces and Invariant Affine Connections,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 10, 23–31 (1998) [Russian Mathematics (Iz. VUZ) 42 (10), 21–29 (1998)].
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Original Russian Text © Yu.D. Churbanov, 2008, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2008, No. 8, pp. 43–57.
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Churbanov, Y.D. Integrability of canonic affinor structures of homogeneous periodic Φ-spaces. Russ Math. 52, 35–47 (2008). https://doi.org/10.3103/S1066369X08080057
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DOI: https://doi.org/10.3103/S1066369X08080057