Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 26 / 5 / b / Solution

Identify the ambient Euclidean space ${\mathbb{R}}^{2m+2}$ with ${\mathbb{C}}^{m+1}$ and let $J$ be multiplication by $i$. Equivalently, in real coordinates,This linear map preserves the norm and satisfies $⟨x,Jx⟩=0$. On the unit sphere $S^{2m+1}$ defineIt is continuous, tangent because it lies in $x^{\perp }=T_x{S}^{2m+1}$, and has norm one everywhere. It is therefore a nowhere-zero section of the tangent bundle, including when $m=0$ and the sphere is $S^1$. This supplies a nonvanishing tangent field on an odd-dimensional sphere explicitly.