On affine Riemann surfaces

We show that the universal covering space of a connected component of a regular level set of a smooth complex valued function on \({\mathbb{C}}^2\), which is a smooth affine Riemann surface, is \({\mathbb{R} }^2\). This implies that the orbit space of the action of the covering group on \({\mathbb{R} }^2\) is the original affine Riemann surface.