(compact, non-empty boundary )Surface Geodesics on Hyperbolic Geometry I have a basic knowledge of hyperbolic geometry . I am trying to understand the meaning of "a compact surface S  with non-empty boundary(which is neither a disk nor an annulus )with a complete hyperbolic metric with geodesic boundary" . 
I know about the upper-half plane model with the metric $$ ds^2 =\frac {dx^2+dy^2}{y^2} $$
in which the geodesics are half-circles with the endpoints in the Real line and vertical half-lines with an endpoint on the Real line. But I don't have a clear idea about the issue of geodesics in a surface with a hyperbolic metric. I imagine we may use charts to pullback the metric locally from $ \mathbb R^2 $ and then somehow patch it up with partitions of unity (which should have no problem working because of the compactness of S ). I guess completeness then follows from compactness of S. Could someone please give me some intro. comments and refs? 
Thanks.  
 A: Let's start with generalities. Suppose that $M$ is a smooth manifold with boundary. In order to define the notion of a Riemannian metric on $M$, embed $M$ as a codimension 0 submanifold in a smooth manifold $M'$ without boundary (this is always possible). Now, a Riemannian metric $g$ on $M$ is the restriction of a Riemannian metric $g'$ on $M'$. The manifold $(M,g)$ has totally geodesic boundary iff the boundary of $M$ is totally geodesic in $(M',g')$. 
One say that $(M,g)$ has negative curvature iff the curvature of $(M',g')$ is negative at each point of $M$. Same for "constant curvature". 
Now, in dimension 2, you can use Gauss-Bonnet to show that if $(M,g)$ is compact negatively curved with geodesic boundary, then $\chi(M)<0$. This rules out disks and annuli. Conversely, given  a compact surface $M$ with $\chi(M)<0$, $M$ admits a metric of constant negative curvature with geodesic boundary. You can prove it by gluing "pairs of pants" with geodesic boundary and equal boundary lengths.  
Here are two books as references: 


*

*W.P.Thurston "Three-dimensional Geometry and Topology", section 4.6. 

*W.Abikoff "Real-analytic theory of Teichmuller space", chapter 2, section 3.  
