In "Supersymmetric Field Theories on Three-Manifolds", the metric for a general circle fibration over a Riemann surface $\Sigma$ (a Seifert manifold) is given on page 5 by $$ ds^2=\Omega(z,\bar{z})(d\psi + a(z,\bar{z})dz+ \bar{a}(z,\bar{z})d\bar{z})^2 + c(z,\bar{z})dzd\bar{z}. $$
My question is, what kind of manifolds are described by metrics where $a(z,\bar{z})=\bar{a}(z,\bar{z})=0$? It seems that these are essentially manifolds which are related to $S^1\times \Sigma$ by a local rescaling. But are there familiar examples of manifolds, such as $S^3$, that can be described by such metrics?