Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 25I / c / Solution

Let $M$ be a compact oriented surface with piecewise smooth, positively oriented boundary. If ${\theta }_j$ are the signed exterior turning angles at its corners, the global Gauss-Bonnet theorem with boundary isFor a smooth boundary the corner sum is absent, and for a closed surface both boundary terms are absent.