I'm really struggling with the very basics definitions of this course on Riemannian geometry and I was wondering if someone could point me in the right direction in regard to this question (Please don't just tell me the answer)
I think I mainly know how to do part (a) because identifying the antipodes of the sphere is basically just enforcing that $\forall x \in S^2:x\text{ ~ }-x $ so I can just calculate: $\gamma'(0)(f) = (f \circ \gamma)'(0) = (f \circ c)'(0) = 6$
For part b, I have absolutely no idea because all I know is that $\dfrac{\partial}{\partial x_1}\vert_{\gamma(t)}$ is supposedly the directional derivative (as an operator?) along the $x_1$ coordinate.
My thought is that I should compose $\phi $ with $c$ because I guess by avoiding $z_1 = 0$ you can just think about a semiphere without its boundary
but I really have absolutely no clue how to begin writing this in the desired form. Please, I'd greatly appreciate just any tips or an explanation for what is actually going on.
EDIT: I'm now thinking that:
$D\phi(\gamma(t))\gamma'(t) = (\phi \circ \gamma)'(t) = \gamma'(t) \in T_{\phi(\gamma(t))}\mathbb{R}P^2$ which sort of looks like what I want. So I'm thinking I should precompose by $\phi$ and differentiate and the two partial derivatives will be just $(1,0)$ and $(0,1)$ and the coefficients will be the differentiated entries
Hints:
Step 1. Compute $c'(t)$ for general $t$ as a function of $t$.
Step 2. Compute $h=\varphi\circ \pi$.
Step 3. Compute both $x_1, x_2$ -components of the vector field (along $\varphi\circ \gamma$) $Dh(c'(t))$ as functions of $t$.
Step 4. These will be $\alpha_1(t)$ and $\alpha_2(t)$.