Proposition: Regular flatness and nontrivial gluing

The metric (eq:metric) is locally flat on $M$. In the convention

the gluing map $J_{\mu }$ fixes $T,z$ and rotates the developed $(x,y)$-plane by $\mu$. Thus local flatness does not remove the global conical gluing.

Proof. Differentiating (eq:development) gives

For the deck transformation, the developed spatial angle changes by $-2\pi \alpha =\mu -2\pi$, which is the same rotation as $\mu$.