**َ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.



An equivalent definition for contact pair manifolds
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.


