Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 115 / 3 / b

Solution

S^{n} \times S^{n}

, consider

q (x, y) = ⟨ x, y ⟩

. At a point of

q^{- 1} (0)

, its differential is

d q_{(x, y)} (u, v) = ⟨ u, y ⟩ + ⟨ x, v ⟩ .

(y, x)

lies in

T_{x} S^{n} \oplus T_{y} S^{n}

and has

d q (y, x) = 2

P = q^{- 1} (0)

T_{(x, y)} P = {(u, v) : ⟨ u, x ⟩ = 0, ⟨ v, y ⟩ = 0, ⟨ u, y ⟩ + ⟨ x, v ⟩ = 0} .

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