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

On $S^n\times S^n$, consider $q(x,y)=⟨x,y⟩$. At a point of $q^{-1}(0)$, its differential isThe tangent vector $(y,x)$ lies in $T_x{S}^n\oplus T_y{S}^n$ and has $dq(y,x)=2$, so zero is a regular value. The regular level set theorem makes $P=q^{-1}(0)$ a smooth submanifold. Its tangent space is