CONHARMONIC ANALOGUS OF GRAY IDENTITIES FOR ALMOST CONTACT METRIC MANIFOLDS OF THE CLASS
The conharmonic analogues of Gray identities for one of the most interesting class of almost contact metric manifolds are generalized, which generalizes the class of cosymplectic manifolds. In the Chinea-Gonzales classification this class of manifolds is called a class. The Ricci tensor of С <sub>10</sub> -manifolds is Φ-invariant. On the basis of contact analogues of Gray identities, three classes of С <sub>10</sub> -manifolds are distinguished for the harmonic curvature tensor that satisfy these identities and are denoted by. In this paper it is proved that a С <sub>10</sub> -manifold is a -manifold. It is proved that the necessary and sufficient condition for a С <sub>10</sub> -manifold to be a -manifold is that the structure tensor vanishes. A local structure of С <sub>10</sub> -manifolds of classes and is obtained. A local structure of conharmonic flat С <sub>10</sub> -manifolds is obtained. An example of such a manifold is a product. Identities are obtained for the tensor of conharmonic curvature under which the С <sub>10</sub> -manifolds are of class and. It is proved that a С <sub>10</sub> -manifold of a class is an Einstein manifold with a cosmological constant. A С <sub>10</sub> -manifold of a class of dimension greater than 3 has zero scalar curvature. It is proved that a -manifold of dimension greater than 5 is of class if and only if it is a flat cosymplectic manifold.