 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-115/1/c/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

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

choose the outward unit normal

N (x) = x

X, Y

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

A (X, Y) = - ⟨ X, Y ⟩ N .

X, Y

are orthonormal, the Gauss equation gives

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

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

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