Sasaki structures on general contact manifolds

Abstract

We extend the notion of a Sasakian structure from the classical setting of a cooriented contact manifold, where it is given by a compatibility between a contact form $\hat{I}\cdot$ and a Riemannian metric $g_M$ on $M$, to the case of an arbitrary contact structure understood as a contact distribution. In the cooriented case, this compatibility can be equivalently expressed by the fact that the symplectic form $\ddot{I}‰=\mathrm{d}(s^2\hat{I}\cdot )$ and the cone metric $g(x,s)=\mathrm{d}s\otimes \mathrm{d}s+s^2{g}_M(x)$ define a Kähler structure on the cone $\mathcal{M}=M\times {\mathbb{R}}_{+}$. Since general contact structures admit canonical realizations as homogeneous symplectic structures $\ddot{I}‰$ on principal ${\mathbb{R}}^{\times }$-bundles $P\to M$, it is natural to interpret Sasakian geometry in full generality in terms of suitable homogeneous Kähler structures on $P$. We characterize homogeneous Kähler structures on symplectizations $(P,\ddot{I}‰)$ associated with arbitrary contact structures on $M$, and show that they canonically determine a two-sheeted covering $\tilde{M}$ of $M$ equipped with a contact form. This reduces the problem to the cooriented case and leads to a notion of a generalized Sasakian structure on $M$ associated with a homogeneous Kähler structure on $(P,\ddot{I}‰)$. Moreover, since products of Kähler manifolds are again Kähler, our framework naturally yields a concept of a product of Sasakian manifolds. The whole constructions are intrinsic and conceptual, avoiding any ad hoc choices.

35 pages, corrected and substantially rewritten