Constant mean curvature spheres in homogeneous three-manifolds

We prove that two spheres of the same constant mean curvature in an arbitrary homogeneous three-manifold only differ by an ambient isometry, and we determine the values of the mean curvature for which such spheres exist. This gives a complete classification of immersed constant mean curvature spheres in three-dimensional homogeneous manifolds.

Naturally reductive homogeneous (α,β) spaces

In this paper, we study naturally reductive [Formula: see text]-metrics on homogeneous manifolds. We show that naturally reductive [Formula: see text]-metrics arise only when [Formula: see text] is naturally reductive and some conditions on [Formula: see text] is satisfied. We give an explicit formula for the flag curvature of naturally reductive [Formula: see text]metrics which improves the flag curvature formula of naturally reductive Randers metrics given in [D. Latifi, Naturally reductive homogeneous Randers spaces, J. Geom. Phys. 60 (2010) 1968–1973]. As a special case, we give an explicit formula for the flag curvature of bi-invariant [Formula: see text]-metrics on Lie groups.