 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-4/25i/c/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Let

M

be a compact oriented surface with piecewise smooth, positively oriented boundary. If

θ_{j}

are the signed exterior turning angles at its corners, the global Gauss-Bonnet theorem with boundary is

\int_{M} K d A + \int_{\partial M} κ_{g} d s + j \sum θ_{j} = 2 π χ (M) .

For a smooth boundary the corner sum is absent, and for a closed surface both boundary terms are absent.

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