Topological point of view:
in topological point of view a flat surface is a topological space $ M$ with a (ramified in nontrivial case) map$ \pi:M\longrightarrow T^2$.
and the map satisfied:$ \pi$ is not ramified on $ \pi^{-1}(T^2-\{0\})$ is not ramified and defined a covering map.
Geometric-analytic point of view:
we begin with compact connected oriented surface $ M$,and a nonempty finite subset $ \Sigma=\{A_1,…,A_n\}$ of M.and
translation surface of type $ k$:
translation structure:
complex structure: