Flat surface 1

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: