Abstract
We characterize Riemannian orbifolds and their coverings in terms of metric geometry. In particular, we show that the metric double of a Riemannian orbifold along the closure of its codimension one stratum is a Riemannian orbifold and that the natural projection is an orbifold covering.
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Acknowledgements
I would like to thank Claudio Gorodski, Alexander Lytchak, Ricardo Mendes and Marco Radeschi for useful discussions. I am moreover grateful to the referee for comments that helped to improve the exposition. The author was partially supported by the DFG funded project SFB/TRR 191. The support is gratefully acknowledged.
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The author was partially supported by the DFG funded Project SFB/TRR 191.
Appendix
Appendix
To fill in the details for our claim that the map \(\theta : 2\mathcal {O}\supset B_{r/9}(x) \rightarrow B_{r/9}(y) \subset M/H\) in the proof of Proposition 1.3 is an isometry, we first state some simple facts about the metric double of a metric space (X, d) along a closed subspace Y. In the following we denote the two copies of X sitting in \(2_Y X\) by \(X_1\) and \(X_2\).
Lemma 5.1
Let \(\gamma :[0,1] \rightarrow 2_Y X\) be a path connecting points \(\gamma (0) \in X_1\) and \(\gamma (1) \in X_2\). Then the set \(M=\{t\in [0,1]|\gamma (t) \in Y\}\subset [0,1]\) is a nonempty union of closed intervals.
Proof
The subspaces \(X_1\) and \(X_2\) are closed in \(2_Y X\) since Y is closed in X. Moreover, we have \(2_Y X = X_1 \cup X_2\) and \(Y=X_1 \cap X_2\). Hence, \([0,1]=\gamma ^{-1}(X_1)\cup \gamma ^{-1}(X_2)\) and so there is some \(t \in [0,1]\) with \(\gamma (t) \in X_1 \cap X_2 =Y\) since [0, 1] is connected. This shows that M is non-empty. By continuity M is also closed. Every closed subset of [0, 1] is a union of closed intervals. \(\square \)
There is a natural reflection \(s:2_Y X \rightarrow 2_Y X\) that interchanges the two copies \((X_1,d)\) and \((X_2,d)\) of X in \((2_Y X,d_2)\) and fixes the subspace \(Y=X_1\cap X_2\subset 2_Y X\) pointwise. The fact that it identifies \(X_1\) and \(X_2\) isometrically by definition implies that it is an isometry.
Lemma 5.2
For two points \(x,x' \in X_1\) we have \(d(x,x')=d_2(x,x')\le d_2(x,s(x'))\). In other words, the embedding \(X_1 \hookrightarrow 2_Y X\) is isometric and the composition with the projection from \(2_Y X\) to \(2_Y X/s\) with the quotient metric is an isometry.
Proof
For any approximation \(\sum _{i=0}^{k-1} d(x_i,x_{i+1})\) of \(d_2(x,x')\) or \(d_2(x,s(x'))\) we obtain the same approximation of \(d(x,x')\) by mapping all the \(x_i\) that lie in \(X_2\) to \(X_1\) via s. This shows \(d_2(x,x')\le d_2(x,s(x'))\), and \(d(x,x')\le d_2(x,x')\) by the triangle inequality. On the other hand, we have \(d_2 \le d\) by the characterization of \(d_2\) as a maximal metric that is majorized by d and so the claim follows. \(\square \)
To prove that \(\theta : 2\mathcal {O}\supset B_{r/9}(x) \rightarrow B_{r/9}(y) \subset M/H\) is an isometry we show that the metrics on \( B_{r/9}(x)\) and \(B_{r/9}(y)\) satisfy the following properties. Let \(s_G\) be the generator of G/H acting on M/H. We denote the metrics on \(2 \mathcal {O}\) and on M/H by \(d_2\) and \(d_q\), respectively. In the subsequent lemma \((Z,z,d_Z,s_Z,\phi )\) may either be \((2\mathcal {O}, x,d_2,s, \mathrm {id})\) or \((M/H,y,d_q,s_G,\theta )\).
Lemma 5.3
Let \((Z,d_Z)\) be a length space with an isometric involution \(s_Z\). Suppose that there exists a homeomorphism \(\phi : 2 \mathcal {O}\supset B_r(x)\rightarrow B_r(z)\subset Z\) with \(\phi (x)=z\in Z_0:=\mathrm {Fix}(s_Z)\) that is \(\mathbb {Z}_2\)-equivariant with respect to the action of s on \(B_r(x)\) and the action of \(s_Z\) on \(B_r(0)\), and that is a radial isometry in the sense that \(\phi (B_{r'}(x))=B_{r'}(z)\) holds for all \(r'\in [0,r]\). Then the following holds true.
-
(i)
For \(w,w' \in \phi (B_{r/3}(x)\cap X_1)\) we have
$$\begin{aligned} d_Z(w,w')\le d_Z(w,s_Z(w')). \end{aligned}$$ -
(ii)
For \(w\in \phi (B_{r/3}(z)\cap X_1)\) and \(w' \in \phi (B_{r/3}(z)\cap X_2)\) we have
$$\begin{aligned} d_Z(w,w')= \inf _{y'\in Z_0} \left( d_Z(w,y')+d_Z(y',w') \right) . \end{aligned}$$
Proof
(i) Let \(\gamma :[0,1] \rightarrow Z\) be a path connecting w and \(s_Z(w')\) whose length approximates \(d_Z(w,s_Z(w'))\) up to some small \(\varepsilon >0\). Because of \(w,s_Z(w') \in B_{r/3}(z)\), we can assume that \(\gamma \) is completely contained in \(B_{r}(z)\). By Lemma 5.1 applied to \(\phi ^{-1}\circ \gamma \) there is some \(t_0 \in [0,1]\) with \(\gamma (t_0) \in \phi (Y\cap B_{r}(x)) \subset \mathrm {Fix}(s_Z)\). We define a new path \(\tilde{\gamma }:[0,1]\rightarrow Z\) by \(\tilde{\gamma }(t)=\gamma (t)\) for \(t\in [0,t_0]\) and \(\tilde{\gamma }(t)=s_Z(\gamma (t))\) for \(t\in [t_0,1]\). The path \(\tilde{\gamma }\) connects w and \(w'\) and has length \(L(\tilde{\gamma })=L(\gamma )\) since s is an isometry. Now the claim follows since Z is a length space.
(ii) By the triangle inequality we have \(d_Z(w,w') \le \inf _{y'\in Z_0} \left( d_Z(w,y')+d_Z(y',w') \right) \). On the other hand, let \(\gamma \) be a path connecting w and \(w'\) whose length approximates \(d_Z(w,w')\). As above we can assume that \(\gamma \) is completely contained in \(B_{r}(z)\). Similarly as in (i) we can use Lemma 5.1 to construct a path \(\tilde{\gamma }\) connecting w and \(s_Z(w')\) that lies completely in \(\theta (B_{r}(z)\cap X_1)\), intersects \(Z_0\) and satisfies \(L(\tilde{\gamma })=L(\gamma )\). Since \(\tilde{\gamma }\) intersects \(Z_0\), we have \(L(\tilde{\gamma })\ge \inf _{y'\in Z_0} \left( d(w,y')+d(y',w') \right) \). The fact that Z is a length space implies \(d(w,w') \ge \inf _{y'\in Z_0} \left( d(w,y')+d(y',w') \right) \) and hence the claim follows. \(\square \)
Lemma 5.4
The map \(\theta : 2\mathcal {O}\supset B_{r/9}(x) \rightarrow B_{r/9}(y) \subset M/H\) is an isometry as claimed in the proof of Proposition 1.3.
Proof
Let \(z,z' \in B_{r/3}(x)\cap X_1\). By Lemma 5.2 we have \(d(z,z')=d_2(z,z')=d(\overline{z},\overline{z}')\) where \(\overline{z}\) and \(\overline{z}'\) are the cosets of z and \(z'\) in \(2 \mathcal {O}/s\cong \mathcal {O}\). Since \(\theta \) descends to an isometry on \(B_r(x)\subset \mathcal {O}\), the definition of the quotient metric on M/G implies
Now Lemma 5.3, (i), shows that \(d_2(z,z')=d_q(\theta (z),\theta (z'))\). By the same reason this identity holds for points \(z,z' \in B_{r/3}(x)\cap X_2\).
Now let \(z \in B_{r/9}(x)\cap X_1\) and \(z' \in B_{r/9}(x)\cap X_2\). Then we have \(d_2(z,z'),d_q(\theta (z),\theta (z'))\le 2r/9\). Applying the first paragraph and Lemma 5.3, (ii), in \(B_r(x)\) and in \(B_r(y)\) yields \(d_2(z,z')=d_q(\theta (z),\theta (z'))\)
which proves the claim. \(\square \)
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Lange, C. Orbifolds from a metric viewpoint. Geom Dedicata 209, 43–57 (2020). https://doi.org/10.1007/s10711-020-00521-x
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DOI: https://doi.org/10.1007/s10711-020-00521-x