  We introduce the notion of contact pair structure and the corresponding
associated metrics, in the same spirit of the geometry of almost contact
structures. We prove that, with respect to these metrics, the integral curves
of the Reeb vector fields are geodesics and that the leaves of the Reeb action
are totally geodesic. Mreover, we show that, in the case of a metric contact
pair with decomposable endomorphism, the characteristic foliations are
orthogonal and their leaves carry induced contact metric structures.

Differential Geometry