Does spherical symmetry imply spherical submanifolds?

  • #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?
 

Answers and Replies

Related Threads on Does spherical symmetry imply spherical submanifolds?

  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
2
Views
3K
  • Last Post
Replies
2
Views
6K
  • Last Post
Replies
1
Views
11K
  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
2
Views
6K
Replies
4
Views
4K
  • Last Post
Replies
2
Views
2K
Top