On the sasaki-hsu contact structure of the brieskorn manifolds

An odd dimensional real submanifold of a complex analytic manifold which is an embedded submanifold locally defined by some fixed number of analytic equations and one differentiable equation is called a quasianalytic submanifold. We show that, whenever the ambient manifold is Kahler and with exact fundamental form, the quasi-analytic submanifolds have a contact structure which, under some supplementary conditions is Sasakian. The application of this result to the Brieskorn manifolds gives the contact structure of Sasaki and Hsu [5]. In [5], S. Sasaki and C. J. Hsu constructed a differential 1-form on the so-called Brieskorn manifolds and proved, by a laborious computation, that this form defines a contact structure. In the present note, we shall give a simpler proof of this fact. Moreover, our proof holds for a larger class of manifolds. Also, we shall see that, for some of the Brieskorn manifolds, the respective structure is metric and normal [1]. This is in accordance with the last remark of [5].