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I now about Killing vector fields in GR but recently stumbled upon the notion of "Killing leaves" and couldn't find any simple explanation of this notion. For example, this paper writes: "integral submanifolds generated by vector fields of a Killing algebra are called Killing leaves." What exactly are Killing leaves and why are they important?

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    $\begingroup$ Minor comment to the post (v2): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. $\endgroup$ – Qmechanic Feb 4 at 17:22
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On page 3, it's defined as:

integral submanifolds of the distribution, generated by vector fields of a Killing algebra $\mathcal{G}$, are called Killing leaves,

A good overview of a (tangent) distribution and how it relates to the foliation of a manifold into "leaves" (hypersurfaces) might be found here, or on notes about the Frobenius theorem, or John Armstrong's blogpost.

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