In Carroll's "spacetime and geometry" he defines a spherical symmetrical spacetime as a spacetime $(M,g)$ for which there exists a Lie algebra homomorphism between the Lie algebra of a subset of the killing vectors of g and the Lie algebra of $SO(3)$.
Now, this does imply by Frobenius theorem that there exists a foliation of 2-dimensional submanifolds of M, however how can we conclude, as Carroll does, that these submanifolds are actually spheres $S^2$?
In order to do so we would need that he above statement of spherical symmetry implies that there exists diffeomorphisms from the foliating submanifolds (leaves) to $S^2$. However, these diffeomorphisms need to be homeomorphisms which implies the submanifolds must also have a topology of spheres.
Does spherical symmetry as defined above actually imply all these things? If not, in what sense are the submanifolds spheres?