"A manifold (with a metric tensor) is said to be spherically symmetric iff the Lie algebra of its Killing vector fields has a sub-algebra that is the Lie algebra of SO(3)." Why? The statement is paraphrased from texts such as Schutz or D'Inverno, where it is always expressed like a definition with no explanation.. except to note that this definition is sufficient to calculate the general form of a spherically symmetric metric/line-element (rather than just writing one down intuitively), which then might be used (in combined with the vacuum-EFE) to obtain Schwarzschild's solution. Could anyone point me to an explanation or proof for the statement? It does make sense that there should be some relationship between the space-time symmetries and the familiar rotation group, but why do we know the relationship to have this specific form?