Differential Geometry of Manifolds

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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.


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