Maximally symmetric subspaces

[Ray]

"Is it possible to derive the Schwarzschild metric from Killing vectors, thus saving all that work with the Ricci tensor etc."

 

[gammamcc]

Given Killing vectors may not determine the metric uniquely. For example, in a specific static case with three Killing vectors on S^2 one still needs the weak field limit imposed on it to recover the metric precisely.

cf Chapt. 7 of
http://pancake.uchicago.edu/~carroll/notes/