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

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- Solution c

Solution

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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