## On a contact 3-structure.(English)Zbl 1004.53058

If a 1-form $$\eta$$ defines a contact structure on an odd-dimensional manifold $$M$$ and $$g$$ is a Riemannian metric, $$(\eta,g)$$ is said to be a contact metric structure if $$\eta$$ has unit length and the $$(1,1)$$-tensor $$\phi$$, defined by $$g(X,\phi Y)={1\over 2}d\eta(X,Y)$$, satisfies the relation $$\phi^2=-I + \eta\otimes \eta^\sharp$$. Following Sasaki, the author considers the 2-form $$\Omega={1\over 2} d\eta + \eta\wedge dt$$ on $$\widetilde{M}=M\times{\mathbb R}$$ and the almost complex structure $$J$$ defined by $$h(X,JY)=\Omega(X,Y)$$, where $$h$$ is the canonical product metric on $$\widetilde{M}$$. The contact metric structure is Sasakian if $$J$$ is integrable. A set of three contact metric structures $$(\eta_a,g)$$, satisfying a certain compatibility condition is called a contact 3-structure.
This short note is about proving that every contact 3-structure is a Sasakian 3-structure. The proof relies on a lemma of N. Hitchin, concerning the integrability of three almost complex structures, compatible with the same metric, on a $$4m$$-dimensional manifold.

### MSC:

 53D10 Contact manifolds (general theory) 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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