= Isometry group of a Riemannian quotient
{title2=$\operatorname{Isom}(X/\Gamma)=N_{\operatorname{Isom}(X)}(\Gamma)/\Gamma$}
For a complete simply connected <Riemannian manifold> $X$ and a free properly discontinuous deck group $\Gamma$, each isometry of $X/\Gamma$ lifts to an isometry of $X$ normalizing $\Gamma$. Conversely such a normalizing isometry descends; two lifts differ by a deck transformation. This proves the formula, including orientation-reversing isometries if the full normalizer is used.
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