Geometry and the additional smooth structure

These $x$-coordinates are not the developing coordinates $(r\cos (\alpha \phi ),r\sin (\alpha \phi ))$. Assumption ass:completion is additional data. For a quotient angle $\alpha =1/N$, it is not automatically the orbifold test-function structure of the covering plane.

Write

Thus, as a generalized function relative to $\mathrm{d}\Omega$, ${\delta }_{\mathcal{W}}={\alpha }^{-1}{{\delta }^{(2)}}_{\mathrm{Leb}}(x)$. Derivatives are transposed with respect to the constant density (eq:density). Put

Indices $A,B$ range over $t,z$, and $i,j$ over $1,2$.

Let $ϵ$ be the positive quarter-turn matrix,

The smooth vector $R$ is ${\partial }_{\phi }$ on $M$.