I'm trying to solve this differential geometry exercise:
Show that a vector field on a $n$-sphere and a smooth map $\Phi: S^n\to \mathbb{R}^{n+1}$ such that $\Phi(x)$ is always orthogonal to $x$ are essentially the same object.
(The $n$-sphere is taken with its standard differentiable structure, given by the two stereographic charts $U,V$)
I think that this exercise requires us to find a bijection between the set of vector fields on $S^n$: $$\text{Der}(C^\infty (S^n))$$ and the set of smooth maps that satisfy that orthogonality property.
I read a solution that went like this. On a stereographic chart, a vector field $X$ looks like this:
$$X|_U=\sum_i \varphi_i \frac{\partial}{\partial x_i}$$
Evaluating this vector field on the function $\sum_{i=1}^n x_i^2$, we get:
$$\sum_{i=1}^n \varphi_i(x_1,...,x_n)x_i=0$$
So we just need to take $\Phi=(\varphi_1,...,\varphi_n)$.
But this doesn't seem rigorous. The functions $\varphi_i$ are defined only on $U$ and not on the whole sphere, so we need to account in someway for the missing point.
From $\Phi(x)$ we can construct a vector field in $\mathbb R^{n+1}$ (where we indentify $T_x\mathbb R^{n+1}$ with $\mathbb R^{n+1}$ ): $$X(x)=|x|\Phi\left(\frac{x}{|x|}\right). $$ Now i use the following fact: $X$ is a vector field in $\mathbb R^{n+1}$ such that $X(F)=0$ then $X_{|M}$ it's a vector field on M (where $M=F^{-1}(0)$ and $0$ is regular value).
In our case we have $F=\sum_{i=1}^{n+1}x_i^2-1$: $$X(F)(x)=|x|\sum_{i=1}^{n+1}\Phi_i\left(\frac{x}{|x|}\right)\frac{\partial}{\partial x_i}(F)= $$ $$=|x|\sum_{i=1}^{n+1}\Phi_i\left(\frac{x}{|x|}\right)2x_i =2|x|\left\langle \Phi\left(\frac{x}{|x|}\right),x\right\rangle=0$$