Differential Geometry of Manifolds

2022 №53

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A note on Gray problem



We consider posed in 1960s Alfred Gray problem on the existence of a six-dimensional non-Kählerian almost Kählerian manifold.

We study six-dimensional almost Hermitian locally symmetric sub­manifolds of Ricci type of Cayley algebra (the notion of such six-dimensional submanifolds of the octave algebra was introduced by Vadim Feodorovich Kirichenko).

Our main result is the following: it is proved that a six-dimensional almost Hermitian locally symmetric submanifold of Ricci type of Cayley algebra does not admit a non-Kählerian almost Kählerian structure.


1. Gray, A., Hervella, L. M.: The sixteen classes of almost Hermitian ma­nifolds and their linear invariants. Ann. Mat. Pura Appl., 123:4, 35—58 (1980).

2. Gray, A.: Some examples of almost Hermitian manifolds. Illinois J. Math., 10:2, 353—366 (1966).

3. Armstrong, J.: An ansatz for almost-Kähler, Einstein 4-manifolds. J. für die Reine und Angewandte Mathematik, 542, 53—84 (2002).

4. Banaru, M.: Geometry of 6-dimensional Hermitian manifolds of the octave algebra. J. Math. Sci. (New York). 207:3, 354—388 (2015).

5. Kirichenko, V. F.: Hermitian geometry of six-dimensional symmet­ric submanifolds of the Cayley algebra. Moscow University Math. Bull. Ser. 1. Mat. Mekh., 3, 6—13 (1994).

6. Banaru, M. B.: On locally symmetric 6-dimensional Hermitian sub­manifolds of Cayley algebra. DGMF. Kaliningrad. 47, 11—17 (2016).

7. Banaru, M.: On the type number of six-dimensional planar Hermi­tian submanifolds of Cayley algebra. Kyungpook Math. J., 43:1, 27—35 (2003).