Abstract
The structure of nearly Kähler manifolds was studied by Gray in several articles, mainly in Gray (Math Ann 223:233–248, 1976). More recently, a relevant progress on the subject has been done by Nagy. Among other results, he proved that a complete strict nearly Kähler manifold is locally a Riemannian product of homogeneous nearly Kähler spaces, twistor spaces over quaternionic Kähler manifolds and six-dimensional (6D) nearly Kähler manifolds, where the homogeneous nearly Kähler factors are also 3-symmetric spaces. In the present article, we show some further properties relative to the structure of nearly Kähler manifolds and, using the lists of 3-symmetric spaces given by Wolf and Gray, we display the exhaustive list of irreducible simply connected homogeneous strict nearly Kähler manifolds. For such manifolds, we give details relative to the intrinsic torsion and the Riemannian curvature.
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References
Alekseevskii D.V.: Compact quaternion spaces. Funk. Anal. i Prilozen. 2(2), 11–20 (1968)
Besse A.L.: Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3 Folge. Springer, Berlin (1987)
Burstall F.E., Gutt S., Rawnsley J.H.: Twistor spaces for Riemannian symmetric spaces. Math. Ann. 295, 729–743 (1993)
Butruille J.-B.: Classification des variétés approximativement kähleriennes homogènes. Ann. Glob. Anal. Geom. 27, 201–225 (2005)
Butruille J.-B.: Twistors and 3-symmetric spaces. Proc. Lond. Math. Soc. (3) 96, 738–766 (2008)
González-Dávila J.C., Martín Cabrera F.: Harmonic G-structures. Math. Proc. Camb. Philos. Soc. 146, 435–459 (2008)
Gray A.: Riemannian manifolds with geodesic symmetries of order 3. J. Differ. Geom. 7, 343–369 (1972)
Gray A.: The structure of nearly Kähler manifolds. Math. Ann. 223, 233–248 (1976)
Helgason S.: Differential geometry, Lie groups, and symmetric spaces. Academic Press, New York (1978)
Hitchin N.: Kählerian twistor spaces. Proc. Lond. Math. Soc. 43(3), 133–150 (1981)
Kirichenko V.F.: K-spaces of maximal rank. Mat. Zametki 22, 465–476 (1977)
Nagy P.A.: On nearly Kähler geometry. Ann. Glob. Anal. Geom. 22, 167–178 (2002a)
Nagy P.A.: Nearly Kähler geometry and Riemannian foliations. Asian J. Math. 6, 481–504 (2002b)
Nomizu K.: Invariant affine connections on homogeneous spaces. Am. J. Math. 76, 33–65 (1954)
O’Brian N.R., Rawnsley J.H.: Twistor spaces. Ann. Glob. Anal. Geom. 3(1), 29–58 (1985)
Tricerri F., Vanhecke L.: Homogeneous Structures on Riemannian Manifolds, London Math. Soc. Lect. Note Series 83. Cambridge University Press, Cambridge (1983)
Watanabe Y., Takamatsu K.: On a K-space of constant holomorphic sectional curvature. Kodai Math. Semin. Rep. 25, 297–306 (1973)
Wolf J.A., Gray A.: Homogeneous spaces defined by Lie group automorphisms, I, II. J. Differ. Geom. 2, 77–159 (1968)
Yano K.: Differential Geometry on Complex and Almost Complex Spaces. International Series of Monographs in Pure and Applied Mathematics, 49. Pergamon Press, New York (1965)
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Dávila, J.C.G., Cabrera, F.M. Homogeneous nearly Kähler manifolds. Ann Glob Anal Geom 42, 147–170 (2012). https://doi.org/10.1007/s10455-011-9305-x
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DOI: https://doi.org/10.1007/s10455-011-9305-x