# How to determine number of boundaries of a manifold from its metric?

Let there be a three-dimensional manifold $M$, with the metric $$ds^2=\frac{d\sigma^2+e^{A(x,y)}(dx^2 +dy^2)}{\sin^2 \sigma},$$ where $x$ and $y$ are local coordinates on a Riemann surface $\Sigma$, and $e^{A(x,y)}$ is a conformal factor for the metric on $\Sigma$, while $\sigma\in [0,\pi]$.

How can we see that there are two boundaries for $M$ at $0$ and $\pi$? Is this because the metric diverges at these points?

Also, if I restrict $\sigma$ such that $\sigma\in [0,\frac{\pi}{2}]$, will the manifold have only one boundary?

Near the endpoint $\sigma=0$ we have $\sin\sigma\approx\sigma$ so the metric looks like a deformation of the hyperbolic metric $\frac{1}{\sigma^2}(d\sigma^2+dx^2+dy^2)$ on the upperhalfspace (leaving out the conformal factor for simplicity). So actually there is no boundary component near $\sigma=0$ since the hyperbolic metric is complete. Similarly for the other end $\sigma=\pi$. If you restrict to $(0,\frac{\pi}{2}]$ you create a boundary component at $\sigma=\frac{\pi}{2}$.