Translation surface

This article covers translation surfaces in complex geometry. For translation surfaces in the sense of elementary differential geometry, i.e. surfaces that are created by shifting a curve along a guide curve, see sliding surface .

A translation surface is a mathematical object from the sub-area of geometry . There are several equivalent ways of defining this area. In this article the definition by means of maps is chosen.

definition

A translational surface is a coherent, compact orientable surface with sex , a finite, non-empty set of singularities and a two-dimensional Atlas on , so that the card is inserted, illustrations of translations are. ${\displaystyle (X,\Sigma ,\omega )}$ ${\displaystyle X}$ ${\displaystyle g\geq 1}$ ${\displaystyle \Sigma }$ ${\displaystyle \omega }$ ${\displaystyle X^{\ast }:=X\setminus \Sigma }$ ${\displaystyle \omega }$

Equivalently, a translation surface can be defined as a Riemann surface with a holomorphic 1-form (an Abelian differential ). The singularities of the translation surface correspond to the zeros of the 1-form.

Examples

• If you take a square in the plane and glue the opposite sides by means of translations, a torus is created. The torus is thus a translation surface without singularities. The angles around the corner (after gluing there is only one) add up , which is why the image of this corner is not a (or a liftable) singularity. ${\displaystyle 2\pi }$
• A slightly more complicated translation surface is created when two regular pentagons are glued together. If you turn the pentagons so that one of their sides is horizontal, with one pentagon above its horizontal side and the other below, then two sides are parallel. If you glue the parallel sides, a compact surface is created. Here, too, all corners are identified with one another. The result is a conical singularity with an angle (the sum of the interior angles of the corners). In contrast to the torus, the singularity cannot be lifted in this case. The resulting translation surface has gender 1. This can be calculated, for example, with the help of the Euler characteristic . ${\displaystyle 6\,\pi }$
• In general, every translation surface is created from a finite number of Euclidean polygons by identifying pairs of sides through translations. In doing so, oppositely oriented sides are identified with one another and the orientations are selected so that the edge of the polygons lying in the plane is traversed clockwise. The corners of the polygons correspond to (possibly liftable) singularities of the translation surface, the angles around a corner add up to a multiple of . All singularities are images of corners of the polygons. ${\displaystyle 2\pi }$

Holonomy and singularities

A flat metric defines a parallel transport , the holonomy of which is the rotation around the cone angle of the singularity along a closed path encircling a conical singularity. The singularities of a translation surface are conical singularities with a cone angle that is an integral multiple of . Therefore, translation surfaces have trivial holonomy. ${\displaystyle 2\pi }$

application

Translation surfaces can be used, for example, to investigate (frictionless) billiard lanes in rational polygon-shaped billiard tables. Instead of the reflection of the (punctiform) billiard ball on one side of the polygon, the polygon is mirrored on this side and the billiard track continues in a straight line.

Web links

• Howard Masur : Ergodic theory of translation surfaces
• Giovanni Forni , Carlos Matheus : Introduction to Teichmüller theory and its applications to dynamics of interval exchange maps, flows on surfaces and billiards
• Jean-Christophe Yoccoz : Interval exchange maps and translation surfaces
• Anton Zorich : Flat surfaces

Individual evidence

1. ↑ TO Zemlyakov, AB Katok: Topological Transitivity of Billiards in the polygon. In: Mathematical Notes. 18, 2, pp. 760-764.