I'm looking at Lecture Notes on General Relativity, Sean M. Carroll, 1997. I dont understand eq 7.4 from the theorem 7.2. As I understand, theorem 7.2 is used when you have submanifold that foilate the manifold, and the submanifold must be maximally symmetric. I know that 2-spheres are maximally symmetric, and foliate a spherically symmetric 3-space . I don't understand the introduction of 2 coordinates, i.e. applying the therem to 4-d space-time, as the submanifolds only foliate 3-space, I thought the application of the theoremm would be limited to 3-space? Also does thsi theorem have a name? So i can look it up elsewhere? Thanks in advance.