Let *G* be a Lie group and *H* be a compact subgroup. The (left) coset space *G/H* is, up to an isomorphism, equivalent to the smooth homogeneous manifold *M*. My question is, is it possible to impose an explicit Riemannian structure on this manifold by specifying *G* as an isometry group with corresponding isotropy group *H*? For example, consider the group *SO(3)*. The coset space *SO(3)/SO(2)* is then isomorphic to the 2 sphere. If one specifies *SO(3)* as the group of isometries, is it possible to recover an explicit expression for the metric? Would this necessarily be the "traditional" metric inherited from embedding in Euclidean 3D space? Thanks