Does spherical symmetry imply spherical submanifolds?

1. Jun 4, 2014

center o bass

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 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 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?