If one has a certain (semi)Riemannian manifold $(M,g)$ with a Killing vector field $X$, then the flow $\phi$ of $X$, forms a one-parameter isometry group $G$ on $M$. Then one can define an equivalence relation $\sim$ between points in $M$ by $p\sim p'$ iff $p'=h(p)$ for some $h\in G$.

I want to know whether the resulting quotient space $M/\sim$ is a manifold and if so is there a canonical way of defining a metric on $M/\sim$ using the original metric $g$ on $M$?

  • 1
    $\begingroup$ This is a problem in pure mathematics, really. Did you check math.SE first before posting it here? $\endgroup$
    – DanielC
    May 17, 2021 at 23:14
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    $\begingroup$ math.stackexchange.com/q/496571 $\endgroup$
    – G. Smith
    May 17, 2021 at 23:30
  • $\begingroup$ Thank you, and yes It may have been better to ask it there. $\endgroup$ May 17, 2021 at 23:34

1 Answer 1


Locally, this can always be done. That is, the local quotient of a (semi-)Riemannian manifold by the orbit of the isometry group induced by a Killing vector is again locally (semi-)Riemannian. Globally, this isn't necessarily true anymore; instead, the quotient is an orbifold, which might have singularities.

  • $\begingroup$ Thank you, can you suggest any good source for introduction to orbifolds. I am not an expert in the string theory so I would appreciate if it doesn't rely on string theory. $\endgroup$ May 17, 2021 at 23:32
  • $\begingroup$ Nakahara's book on mathematical physics has a tiny section about them. For a more in-depth exposition, the classic text is Orbifolds and String Topology by Adem, Leida, and Ruan. You might want to take a look at Satake's original paper (where he calls orbifolds "V-manifolds") as well. None of these rely on String theory per se, just that they have lots of applications. $\endgroup$
    – jsborne
    May 17, 2021 at 23:50

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