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I've been reading up on Killing vectors, and have got on to the topics of homogeneous, isotropic and maximally symmetric space-times. I've read that for an

*isotropic*spacetime, one can construct a set of Killing vector fields ##K^{(i)}##, such that, at some point ##p\in M## (where ##M## is the space-time manifold), ##K^{(i)\mu}(p)=0## and ##\nabla_{\nu}K^{(i)\mu}(p)\neq 0##. Intuitively, I understand that we must have ##K^{(i)\mu}(p)=0##, since isotropy is a rotational symmetry, and so we don't want the Killing vectors to translate the points, we want them to rotate about them. However, I don't understand why we require ##\nabla_{\nu}K^{(i)\mu}(p)\neq 0##. Why must the covariant derivative of each Killing vector be arbitrary? I read that it has something to do with ##\nabla_{\nu}K^{(i)\mu}(p)## generating "rotations" of neighbouring points about ##p##, but this is unclear to me.
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