Editing Gauss–Bonnet theorem (section)

== Statement ==
Suppose {{mvar|M}} is a [[Compact space|compact]] two-dimensional [[Riemannian manifold]] with boundary {{math|∂''M''}}. Let {{mvar|K}} be the [[Gaussian curvature]] of {{mvar|M}}, and let {{math|''k''<sub>''g''</sub>}} be the [[geodesic curvature]] of {{math|∂''M''}}. Then<ref name="doCarmo1992">{{Cite book |title=Riemannian geometry |last=do Carmo |first=Manfredo Perdigão |date=1992 |publisher=Birkhäuser |isbn=0817634908 |location=Boston |oclc=24667701 |author-link=Manfredo do Carmo}}</ref><ref name="doCarmo1976">{{Cite book |title=Differential geometry of curves and surfaces |last=do Carmo |first=Manfredo Perdigão |date=1976 |publisher=Prentice-Hall |isbn=0132125897 |location=Upper Saddle River, N.J. |oclc=1529515 |author-link=Manfredo do Carmo}}</ref>

:<math>\int_M K\,dA+\int_{\partial M}k_g\,ds=2\pi\chi(M), \, </math>
where {{mvar|dA}} is the [[volume element|element of area]] of the surface, and {{mvar|ds}} is the line element along the boundary of {{mvar|M}}. Here, {{math|''χ''(''M'')}} is the [[Euler characteristic]] of {{mvar|M}}.

If the boundary {{math|∂''M''}} is [[piecewise smooth]], then we interpret the integral {{math|∫<sub>∂''M''</sub> ''k''<sub>''g''</sub> ''ds''}} as the sum of the corresponding integrals along the smooth portions of the boundary, plus the sum of the [[angle]]s by which the smooth portions turn at the corners of the boundary.

Many standard proofs use the theorem of turning tangents, which states roughly that the [[winding number]] of a [[Jordan curve theorem|Jordan curve]] is exactly ±1.<ref name="doCarmo1992" />