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Differential Geometry of Manifolds

2020 №51

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Banaru G. A.

On six-dimensional AH-submanifolds of class W1⊕W2⊕W4 in Cayley algebra

DOI
10.5922/0321-4796-2020-51-1
Pages
7-13
almost Hermitian manifold Gray — Hervella classes al­most contact metric structure quasi-Sasakian structure Cayley algebra

Abstract

Six-dimensional submanifolds of Cayley algebra equipped with an almost Hermitian structure of class W1 W2 W4 defined by means of three-fold vector cross products are considered. As it is known, the class W1 W2 W4 contains all Kählerian, nearly Kählerian, almost Kählerian, locally conformal Kählerian, quasi-Kählerian and Vaisman — Gray manifolds. The Cartan structural equations of the W1 W2 W4 -structure on such six-dimensional submanifolds of the octave algebra are obtained. A criterion in terms of the configuration tensor for an arbitrary almost Hermitian structure on a six-dimensional submanifold of Cayley algebra to belong to the W1 W2 W4 -class is established. It is proved that if a six-dimensional W1 W2 W4 -submanifold of Cayley algebra satisfies the quasi-Sasakian hypersurfaces axiom (i.e. a hypersurface with a quasi-Sasakian structure passes through every point of such submanifold), then it is an almost Kählerian manifold. It is also proved that a six-dimensional W1 W2 W4 -submanifold of Cayley algebra satisfies the eta-quasi-umbilical quasi-Sasakian hypersurfaces axiom, then it is a Kählerian manifold.

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