Consider the map $\hat{G}_{s} = \begin{pmatrix} 1 & 0 & 0\\ 0 & cos(s) & sin(s)\\ 0 & -sin(s) & cos(s) \end{pmatrix} : \mathbb{R}^3 \to \mathbb{R}^3$.
- Prove that $\hat{G}_s$ induces a diffeomorphism $G_s : S^2 \to S^2$ and show that this defines a flow $G$ on $S^2$.
- Find the associated vector field $\zeta ^G$ and find the points where $\zeta ^G$ is zero.
Question 1 is fine, my problem is with Question 2. My attempt:
The associated vector field is $\zeta ^G : S^2 \to TS^2, (x,y,z) \mapsto [\sigma _x]$, where $\sigma_{x}(t) = G_t (x)$. Since $G$ is a flow on $S^2$ we have $\sigma_{x}(0) = G_0 (x) = x$. Now we need to find where $\zeta ^G$ is $0$. i.e. the set $Z$:
$$ Z = \{(x,y,z) \in S^2 : [\sigma _x] = 0\} = \{(x,y,z) \in S^2 : D\sigma_x |_0 = 0 \} \tag{1}$$ and we can calculate $$ D\sigma_x |_0 = (0,z,-y)$$ So $$ Z = \{(x,y,z) \in S^2 : (0,z,-y)=0 \}$$, i.e. the x-axis.
My attempt is apparently incorrect and I have no idea why. My questions:
- Where have I gone wrong?
- Is the equality in (1) true? I think it is because there exists a bijection between $T_x X$ and $\mathbb{R}^n$
I would appreciate any help.