An equivalent definition for contact pair manifolds

Question

َContact manifold

A $(2n + 1)$-dimensional manifold $M$ is said to be a contact manifold if it admits a global 1-form $\eta$ such that $\eta\wedge (d\eta)^n\neq 0$.

There is an equivalent definition by J. W. Gray:

A manifold admit an almost contact structure if the structural group of the tangent bundle is reducible to $U(n)\times 1$.

Contact pairs manifold

A pair $(\alpha_1, \alpha_2)$ of $1$-forms on a manifold is said to be a contact pair of type $(h,k)$ if $$\alpha_1 \wedge (d\alpha_1)^h \wedge\alpha_2 \wedge(d\alpha_2)^k\ \text{ is a volume form,}\\ (d\alpha_1)^{h+1} = 0\ \text{and}\ (d\alpha_2)^{k+1} = 0.$$

Question: Is there an equivalent definition for contact pair similar to contact case?

Thanks.

Answer 1

As Jarek said, the two definitions that you give are definitely NOT equivalent: the definition by Gray is about almost contact structures which is a much weaker notion than a proper contact structure!

Maybe you would be interested in an almost contact pair like $TM$ splits into $\mathbb{R}\oplus E_1\oplus \mathbb{R}\oplus E_2$, where $E_1$ and $E_2$ are complex bundles of (complex) dimensions $h$ and $k$ respectively. Is this useful for anything? I don't know.