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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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