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

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.

Twistorial construction of minimal hypersurfaces

Every almost Hermitian structure (g, J) on a four-manifold M determines a hypersurface ΣJ in the (positive) twistor space of (M, g) consisting of the complex structures anti-commuting with J. In this paper, we find the conditions under which ΣJ is minimal with respect to a natural Riemannian metric on the twistor space in the cases when J is integrable or symplectic. Several examples illustrating the obtained results are also discussed.

CRITICAL HERMITIAN STRUCTURES ON THE PRODUCT OF SASAKIAN MANIFOLDS

Let [Formula: see text] be a compact orientable smooth manifold admitting an almost complex structure and [Formula: see text] for (λ, μ) ∈ ℝ2 - (0, 0) be the functional defined on the space of the almost Hermitian structure [Formula: see text]. We discuss the first variational problem of the functional [Formula: see text] on the space [Formula: see text] and its subspace [Formula: see text] in the case where [Formula: see text] is a product manifold of Sasakian manifolds. Further this paper provides examples of critical Hermitian structures of the functional [Formula: see text] for various (λ, μ).

HARMONIC CONTACT METRIC STRUCTURES AND SUBMERSIONS

We study harmonic almost contact structures in the context of contact metric manifolds, and an analysis is carried out when such a manifold fibres over an almost Hermitian manifold, as exemplified by the Boothby–Wang fibration. Two types of almost contact metric warped products are also studied, relating their harmonicity to that of the almost Hermitian structure on the base or fibre.

CURVATURE AND INTEGRABILITY OF AN ALMOST HERMITIAN STRUCTURE

By using moving frame theory, we obtain some necessary conditions involving curvatures for integrability of an almost Hermitian structure. As consequences, they are applied to S6.