# Most general version for the Gauss-Bonnet theorem for polygons

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

• I assume "Compact, Open, and Connected", means "Open and connected with compact closure"? – Igor Rivin Mar 27 '17 at 18:54
• I don't think you need any additional conditions. You can replace "piecewise geodesic" with "piecewise smooth" as well. – Ryan Budney Mar 27 '17 at 18:56
• @Ryan Budney: Do you know a reference where I can find a proof of your statement? – Sammyy Delbrin Mar 27 '17 at 18:59
• Take the lift of your region $P$ to the orientable 2-sheeted cover, and keep track of corners, curvatures, etc. Put the metric on the cover which makes it a local isometry. – Ryan Budney Mar 27 '17 at 19:11

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".

• Thanks for the reference. I imagined someone had done this work but I would not have known where to look. – Ryan Budney Mar 27 '17 at 21:50
• The main point of this work is that neither the metric nor the curve is assumed to be smooth. – Alexandre Eremenko Mar 28 '17 at 0:00
• Thank you for these references. Both references are very nice. But I could not find a statement of Gauss-Bonnet theorem as general as in my question. For example in the book of Aleksandrov and Zalgaller (english translation) it is assumed that the boundary is simple closed. Can you tell me exactly which result you are referring to? – Sammyy Delbrin Mar 28 '17 at 19:24
• If your region is multiply connected, break it into simply connected pieces by some piecewise geodesics. – Alexandre Eremenko Mar 28 '17 at 21:19

See Pressley's Elementary Differential Geometry book, Chapter 13, to be specific, for an exhaustive discussion.