A new approach to the classification of almost contact metric manifolds via intrinsic endomorphisms

In 1990, D. Chinea and C. Gonzalez gave a classification of almost contact metric manifolds into $2^{12}$ classes, based on the behaviour of the covariant derivative $\nabla^g\Phi$ of the fundamental $2$-form $\Phi$. This large number makes it difficult to deal with this class of manifolds. We propose a new approach to almost contact metric manifolds by introducing two intrinsic endomorphisms $S$ and $h$, which bear their name from the fact that they are, basically, the entities appearing in the intrinsic torsion. We present a new classification scheme for them by providing a simple flowchart based on algebraic conditions involving $S$ and $h$, which then naturally leads to a regrouping of the Chinea-Gonzalez classes, and, in each step, to a further refinement, eventually ending in the single classes. This method allows a more natural exposition and derivation of both known and new results, like a new characterization of almost contact metric manifolds admitting a characteristic connection in terms of intrinsic endomorphisms. We also describe in detail the remarkable (and still very large) subclass of $\mathcal{H}$-parallel almost contact manifolds, defined by the condition $(\nabla^g_X\Phi)(Y,Z)=0$ for all horizontal vector fields, $X,Y,Z\in\mathcal{H}$.