Contact Pairs

We introduce a new geometric structure on differentiable manifolds. A $\mathit{\text{Contact}}$ $\mathit{\text{Pair}}$on a manifold $M$ is a pair $(\alpha ,\eta )$ of Pfaffian forms of constant classes $2k+1$ and $2h+1$ respectively such that $\alpha \wedge d{\alpha }^k\wedge \eta \wedge d{\eta }^h$ is a volume form. Both forms have a characteristic foliation whose leaves are contact manifolds. These foliations are transverse and complementary. Further differential objects are associated to Contact Pairs: two commuting Reeb vector fields, Legendrian curves on $M$ and two Lie brackets on ${\mathcal{C}}^{\mathrm{\infty }}(M)$. We give a local model and several existence theorems on nilpotent Lie groups, nilmanifolds, bundles over the circle and principal torus bundles.