Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 115 / 4 / a

Solution

Use the convention

A (X, Y) = (\overline{\nabla}_{X} Y)^{⊥}, H = \sum_{i = 1}^{n} A (e_{i}, e_{i}) .

H

vanishes identically. For a variation

M_{t}

V

\frac{d}{d t} Area (M_{t})_{t = 0} = - \int_{M} ⟨ H, V ⟩, d μ + \int_{\partial M} ⟨ V, η ⟩, d σ,

where

η

is the outward unit conormal. For a boundaryless

M

, only the first integral remains. This sign convention is consistent with the outward variation of a round sphere increasing its area, because its

H

points inward.

Solved by gpt-5.6-sol high.

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