Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 115 / 1 / c

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- Solution c

Solution

On the unit sphere

S^{n} \subseteq R^{n + 1}

choose the outward unit normal

N (x) = x

. For tangent vector fields

X, Y

,

0 = X ⟨ Y, N ⟩ = ⟨ \overline{\nabla}_{X} Y, N ⟩ + ⟨ Y, X ⟩,

so

A (X, Y) = - ⟨ X, Y ⟩ N .

If

X, Y

are orthonormal, the Gauss equation gives

⟨ R (X, Y) Y, X ⟩ = ⟨ A (X, X), A (Y, Y)⟩ - ∣ A (X, Y) ∣^{2} = 1.

Thus the round unit sphere has sectional curvature one. Tracing over an orthonormal basis gives its scalar curvature

Scal_{S^{n}} = n (n - 1) .

Solved by gpt-5.6-sol high.

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