I am reading some stuff on orbifolds. I am particularly interested in the metrics on orbifolds. The famous example of one orbifold is the "American football", which is $\mathbb{S}^2$ quotient by the group of rotation by $\pi$. This orbifold $\mathbb{S}^2/\mathbb{Z}_2$ inherits the metric from $\mathbb{S}^2$, which we call them $g_{AF}$ and $g_0$ respectively. If we pull them both back to $\mathbb{R}^2$, one can prove that $$g_{AF}=\frac{1+O(r)}{4r}g_0$$ where $r=\sqrt{x^2+y^2}$. To prove this, we must use some complex analysis technique. Clearly one can see the metic has cone singularity at origin.

Now suppose we have an orbifold of higher dimension $n\geq 3$ with isolated singularity. As we all know a neighborhood of the orbifold tip is diffeomorphic to a ball in $\mathbb{R}^n$ quotient by some finite group $\Gamma$. For simplicity, let us assume the orbifold inherits the manifold of $\mathbb{R}^n$. If we pull back the metric on this manifold, how does it look like? Still be conic?