Show that the three vector fields $X = y\frac{\partial}{\partial z}-z\frac{\partial}{\partial y}, Y = z\frac{\partial}{\partial x}-x\frac{\partial}{\partial z}$ and $Z=x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x}$ on $\Bbb R^3$ are tangent to the $2$-sphere $\Bbb S^2$.
I have a definition that a vector field is tangent to submanifold $S \subseteq M$ if for all $p \in S$ the tangent vector $X_p$ is in $T_pS \subseteq T_pM$.
I don't really know how to approach the problem. Why is $X_p$ neccessarily a tangent vector? Using coordinate charts it seems that $X_p$ is of form $$X_p = \sum_{i=1}^n X^i(p) \frac{\partial}{\partial x^i} \bigg|_p$$ but what is this $X^i(p)$ here?
John Lee's book also suggests that $X$ is tangent to a submanifold $S$ if and only if $(Xf) \mid_S = 0$ for every $f \in C^\infty(M)$ such that $f\mid_S \equiv 0$.
Can I use either one of these definitions here?
Best Answer
By Proposition $5.38$ in John Lee's book "Smooth manifolds" (2. Edition), $T_p\mathbb S^2=\ker df_{|p}$, where $f:\mathbb R^3\to\mathbb R$, $(x,y,z)\mapsto x^2+y^2+z^2-1$. Now $\frac 12\mathrm{grad}f(x,y,z)=x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}+z\frac{\partial}{\partial z}$. So for $p=(x,y,z)\in\mathbb S^2$, a tangent vector $v=\alpha\frac{\partial}{\partial x}+\beta\frac{\partial}{\partial y}+\gamma\frac{\partial}{\partial z}\in T_p\mathbb R^3$ is tangent to $\mathbb S^2$ if and only if $\alpha x+\beta y+\gamma z=0$.