Translation surface ($(X,\omega )$)

A surface with an atlas outside finitely many cone points whose changes of coordinate are translations. On a compact surface it is equivalent to a nonzero holomorphic one-form. A zero of order $m$ has cone angle $2\pi (m+1)$.

A square-tiled surface is a connected translation surface built by gluing finitely many equal Euclidean squares along paired parallel edges by translations. After normalizing side length to one, its area is the number of squares. Vertex identifications can create cone singularities with angle $2\pi k$ for a positive integer $k$.

Taking local square coordinates modulo ${\mathbb{Z}}^2$ gives a finite map to the flat torus ${\mathbb{R}}^2/{\mathbb{Z}}^2$. It is a translation covering away from the images of square vertices, with possible branching over the torus origin. Its degree equals the number of squares.

One square with opposite edges identified gives the ordinary flat torus. Two adjacent unit squares forming a $2$-by-$1$ rectangle, with opposite rectangle edges identified, give another flat torus and a degree-two unbranched map to the unit torus. More elaborate vertex gluings produce higher-genus surfaces. This gives a finite combinatorial encoding of examples used in flat geometry and dynamics.

Cut equal straight slits in two translation surfaces of genus one and cross-glue their banks by translation. The connected sum of oriented manifolds has genus two and two cone points of angle $4\pi$, hence two simple zeros of its holomorphic one-form. Choosing one torus of side $\delta$ and a slit of length ${\delta }^2$ creates a flat-small handle without forcing its generator to have small extremal length.