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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?