Richard Schwartz, in *Mostly Surfaces* (Vol. 60. *American Mathematical Soc.*, 2011),
defines (on p.14) a *translation surface* as "a Euclidean cone surface, all of whose 'angle errors' are integer multiples of $\pi$." 
And a *Euclidean cone surface* is "a surface that is flat except at finitely many cone points." The 'angle error' is the *angle deficit*,
$2 \pi - \theta(p)$ at the cone point $p$.


My question is:



> 
> ***Q***. What role does "integer multiples of $\pi$" play in the theory
>  surrounding translation surfaces? Would half-integer multiples of $\pi$
>  be less interesting? Rational multiples of $\pi$?
> 
> 
> 


I ask this without understanding much of the theory of translation surfaces.
Thanks for any insights the more knowledgeable can provide!



Translation surfaces & integer multiples of $\pi$
Translation surfaces are usually only allowed to have integer multiples of $2\pi$ as cone angles. A translation surface (minus the singularities) admits an atlas of charts whose transition functions are all translations, hence the name. This induces a holomorphic 1-form and an oriented foliation on the surface. Note that the holonomy around the cone points of a translation surface is trivial. 


Translation surfaces come up naturally in studying the dynamics of billiards on polygonal tables with corners having rational multiples of $\pi$.


You might enjoy reading Anton Zorich's survey on the subject: <http://arxiv.org/pdf/math/0609392.pdf> 


With integer multiples of $\pi$ at cone singularities you get a half-translation surface, where transition functions are compositions of translations and half-turns. The surface inherits an unoriented foliation and a holomorphic quadratic differential. This sort of structure arises naturally in the solutions to extremal problems on surfaces such as Teichmuller's problem.


