Orbifolds from a metric viewpoint

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.

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

$$\begin{aligned} d_2(z,z')=\min \{d_q(\theta (z),\theta (z')),d_q(\theta (z),s_G(\theta (z')))\}. \end{aligned}$$

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. $\square$