Embeddedness of spheres in homogeneous three-manifolds

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Let X denote a metric Lie group diffeomorphic to R³ that admits an algebraic open book decomposition. In this paper we prove that if Σ is an immersed surface in X whose left invariant Gauss map is a diffeomorphism onto S², then Σ is an embedded sphere. As a consequence, we deduce that any constant mean curvature sphere of index one in X is embedded.