Locally Sasakian manifolds (Q2710858)

A Riemannian manifold \((M,g)\) is said to be Sasakian if the metric cone \((M\times \mathbb{R}_+, \overline g=dr^2 +r^2g)\) is Kähler. Such a manifold admits a canonical Killing vector field and, under some assumptions, the space of leaves of the induced foliation is an orbifold \(Z\). When \(M\) is Einstein-Sasakian, \(Z\) is Kähler-Einstein. In this paper, the authors show that the metric \(g\) of any Sasakian manifold of dimension \(2n+1\) can be locally expressed in terms of a real function \(K\) of \(2n\) variables. This function is a Sasakian analogue of the Kähler potential for the Kähler geometry. It is also shown that the metric \(g\) is Sasakian-Einstein if \(K\) satisfies the Monge-Ampère equations and that every \(2n+1\)-dimensional locally Sasakian-Einstein manifold is generated by a locally Kähler-Einstein manifold of dimension \(2n\). When \((M,g)\) is quasiregular, it fibers in a canonical way over a compact Kähler orbifold \(Z\) and the potential function \(K\) is simply the Kähler potential of the twistor space \(Z\).