Suppose $M$ is a 2-dimensional smooth Riemannian manifold and $P\subset M$ is an open and connected subset with compact closure and a piecewise geodesic boundary. 


My question is: What further conditions must $P$ (and $M$) satisfy such that the Gauss-Bonnet theorem is fulfilled for $P$?


I have found a lot of different versions of the Gauss-Bonnet theorem. All of them demand for instance that $M$ is orientable. Sometimes the boundary of $\partial P$ is supposed to be simple, closed and without cusps. But can't we do better than that? Is there a good survey which treats also more general cases?


Best wishes



Most general version for the Gauss-Bonnet theorem for polygons
There are no other conditions, and in fact a more general statement is true.
The standard reference is the survey of Reshetnyak, Two-dimensional surfaces of bounded curvature,
in the book:


MR1263963 
Geometry. IV. 
Nonregular Riemannian geometry. A translation of Geometry, 4. Translation by E. Primrose. Encyclopaedia of Mathematical Sciences, 70. Springer-Verlag, Berlin, 1993.


The original source is the book of Aleksandrov and Zalgaller, Two-dimensional manifolds of bounded curvature, Trudy Mat. Inst. Steklov. 63 1962. There is an English translation.


Gauss Bonnet is stated and proved there in much larger generality: for "Aleksandrov surfaces of bounded curvature".


