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Does spherical symmetry imply spherical submanifolds?

  1. Jun 4, 2014 #1
    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?
     
  2. jcsd
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