Hyperbolic surface

Question

Let $X=S_1\times S_1−Δ$ , where $Δ=\{(x,y)∈S_1\times S_1|x=y\}$.

I know that this is (one of the) usual model for the cylinder , but how to proof that it is a hyperbolic surface?

Any help is appreciated!

Answer 1

Here's a complete answer:

We say a surface $\Sigma$ admits a hyperbolic metric if there exists a complete, finite area, Riemannian metric on $\Sigma$ of constant curvature $-1$. We then have the following theorem,

Thm. Let $\Sigma$ be any surface (possibly with punctures of boundary). If the Euler characteristic $\chi(\Sigma)<0$ then $\Sigma$ admits a hyperbolic metric. If $\chi(\Sigma)=0$ then $\Sigma$ admits a Euclidean metric.

Since the cylinder $C$ has $\chi(C)=0$ it admits a Euclidean metric.