The only compact orientable $n$-manifolds without boundary which can be given a flat Riemannan structure are tori. I was wondering if we could classify the compact oriented $2$-manifolds with boundary which admit flat Riemannan structures.
Are there any other than the torus and cylinder with disks cut out?
EDIT in response the comments: And disks of course
Every compact surface with (nonempty) boundary admits a flat metric. To get a classification, you need to also impose a condition on the mean curvature of the boundary. I learned all of this from Osgood, Phillips, Sarnak, “Extremals of determinants of Laplacians”.
Here is a construction of a flat metric. Fix a metric $g$ on the compact surface $\Sigma$ with (nonempty) boundary $\partial\Sigma$. From the conformal transformation formula for the Gauss curvature, the metric $e^{2u}g$ will be flat with $e^{2u}g\rvert_{\partial\Sigma} = g\rvert_{\partial\Sigma}$ if and only if
$$ \label{eqn} \tag{$\ast$} \begin{cases} -\Delta u + K = 0 & \text{in $\Sigma$}, \\ u\rvert_{\partial\Sigma} = 0 . \end{cases} $$
Here $\Delta$ is the Laplacian with respect to $g$ and $K$ is the Gauss curvature of $g$. The condition on the boundary is only there to make \eqref{eqn} well-defined.
One easy way to construct a solution of \eqref{eqn} is variationally. Define $\mathcal{F} \colon C^\infty(\Sigma) \to \mathbb{R}$ by
$$ \mathcal{F}(u) := \int_\Sigma \left( \lvert\nabla u\rvert^2 + 2Ku \right) \, \mathrm{dvol}_g . $$
Any minimizer of
$$ \inf \left\{ \mathcal{F}(u) \mathrel{}:\mathrel{} u \in C^\infty(\Sigma) , u\rvert_{\partial\Sigma} = 0 \right\} $$
is necessarily a solution of \eqref{eqn}. The infimum is finite by a combination of Hölder’s inequality and the Poincaré inequality. Thus there is a minimizing sequence which converges to a minimizer of $\mathcal{F}$ in $W_0^{1,2}(\Sigma)$. Since \eqref{eqn} is elliptic, a standard PDE argument implies that the minimizer is in fact smooth.