International Journal of Mathematics and Mathematical Sciences
Volume 32 (2002), Issue 2, Pages 117-127
doi:10.1155/S0161171202006774

The local moduli of Sasakian 3-manifolds

Abstract

The Newman-Penrose-Perjes formalism is applied to Sasakian 3-manifolds and the local form of the metric and contact structure is presented. The local moduli space can be parameterised by a single function of two variables and it is shown that, given any smooth function of two variables, there exists locally a Sasakian structure with scalar curvature equal to this function. The case where the scalar curvature is constant (η-Einstein Sasakian metrics) is completely solved locally. The resulting Sasakian manifolds include S 3, Nil, and SL˜ 2 (ℝ), as well as the Berger spheres. It is also shown that a conformally flat Sasakian 3-manifold is Einstein of positive scalar curvature.