Sasaki structures on general contact manifolds

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 $\eta$ 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 $\omega=\mathrm{d}(s^2\eta)$ and the cone metric $g(x,s)=\mathrm{d} s\otimes\mathrm{d} s+s^2g_M(x)$ define a K\"ahler structure on the cone $\mathcal{M}=M\times\mathbb{R}_+$. Since general contact structures admit canonical realizations as homogeneous symplectic structures $\omega$ on principal $\mathbb{R}^\times$-bundles $P\to M$, it is natural to interpret Sasakian geometry in full generality in terms of suitable homogeneous K\"ahler structures on $P$. We characterize homogeneous K\"ahler structures on symplectizations $(P,\omega)$ 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 K\"ahler structure on $(P,\omega)$. Moreover, since products of K\"ahler manifolds are again K\"ahler, our framework naturally yields a concept of a product of Sasakian manifolds. The whole constructions are intrinsic and conceptual, avoiding any ad hoc choices.