Embeddedness of Spheres in Homogeneous Three-Manifolds

Let $X$ denote a metric Lie group diffeomorphic to $\mathbb{R}^3$ that admits an algebraic open book decomposition. In this paper we prove that if $\Sigma$ is an immersed surface in $X$ whose left invariant Gauss map is a diffeomorphism onto $\mathbb{S}^2$, then $\Sigma$ is an embedded sphere. As a consequence, we deduce that any constant mean curvature sphere of index one in $X$ is embedded.