= 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,
$$
J(x_0,x_1,\ldots,x_{2m},x_{2m+1})=(-x_1,x_0,\ldots,-x_{2m+1},x_{2m}).
$$
This <linear map> preserves the norm and satisfies $\langle x,Jx\rangle=0$. On the unit <sphere> $S^{2m+1}$ define
$$
\boxed{V(x)=Jx.}
$$
It is continuous, tangent because it lies in $x^\perp=T_xS^{2m+1}$, 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.
Back to article page
