On the classification of the almost contact metric manifolds

The vector space of the tensors $\mathcal F$ of type (0,3) having the same symmetries as the covariant derivative of the fundamental form of an almost contact metric manifold is considered. A scheme of decomposition of $\mathcal F$ into orthogonal components which are invariant under the action of $U(n)\times 1$ is given. Using this decomposition there are found 12 natural basic classes of almost contact metric manifolds. The classes of cosymplectic, $\alpha$-Sasakian, $\alpha$-Kenmotsu, etc. manifolds fit nicely to these considerations. On the other hand, many new interesting classes of almost contact metric manifolds arise.