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I am reading the last chapter (Schwarzchild solution and Black Holes) of Sean Caroll's GR notes (http://arxiv.org/abs/gr-qc/9712019).

While talking about spherical symmetry, he says how the Frobenius Theorem can be used to foliate a manifold with spherical symmetry with spheres at each point. This allows us to break down the coordinates of an n dimensional manifold into $u^i$ for the submanifold, and $v^I$ to tell us which submanifold we are on. (If the submanifold being considered in m dimensional, $i$ runs from 1 to m, and $I$ from 1 to n-m) .

I have trouble understanding the claim after this construction i.e.

If the submanifold is maximally symmetric, then it is possible to chose the $u$ co-ordinates such that the metric for the manifold is

$$ds^2=g_{IJ}(v)dv^{I}dv^{J}+f(v)\gamma_{ij}du^i du^j$$

Intuitively how can I see the following:

  1. Why is maximally symmetric a condition? What goes wrong if it is not maximally symmetric?

  2. I understand why $f(v)$ should be just a function of $v$, because if I keep my $v^I$ constant, and traverse on the submanifold associated to that point, the metric should be invariant.

But I don't really understand why $g_{IJ}$ should be only a function of $v$, we are not remaining on the same submanifold while changing $v$.

  1. Why are there no cross terms $dv^I du^j$? Caroll says it is by 'making sure' $\frac{\partial}{\partial v^I}$ are orthogonal to tangent vectors of the submanifold. Can you elaborate on this? Why is this always possible?

I am not looking for detailed mathematical arguments, hand-waving would suffice. But ofcourse, it would be more than wonderful, if both are provided.

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Partial answer:

If $g_{IJ}$ was depending on $u$, this would mean that a displacement at constant $u$, for some displacement $dv$, would give a $ds^2$ depending on $u$.

But this would mean that we may characterize specifically (geometrically) this point $u$ on the submanifold, and this is completely contradictory with the fact that this sub-manifold is maximally symmetric, which means that all points $u$ of the sub-manifold are (geometrically) equivalent.

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