I am taking a course on elementary differential geometry, in which we use Do Carmo "Differential Geometry of Curves and Surfaces" as our textbook. I have handed in a written assignment solving - well, trying to - exersice 4.4.22 in this book, and my solution was not approved. My problem is that I do not understand why my solution is inadequate, and my tutor is unavailable for the time being. I start by sketching my solution, and then I will paraphrase his critique of it.
The problem reads:
Let $S^2={(x,y,z) \in R^3;x^2+y^2+z^2=1}$ and let $p \in S^2$. For each piece-wise parameterized curve $\alpha:[0,l]→S^2$ with $α(0)=α(l)=p$, let $P_{\alpha}:T_p(S^2)→T_p(S^2)$ be the map which assigns to each $v \in T_p(S^2)$ its parallel transport along $α$ back to p. By prop.1, $P_α$ is an isometry. Prove that for every such rotation $R$ of $T_p(S^2)$ there exists an $α$ such that $R=P_α$.
$\textbf{My solution:}$ Let $v \in T_p (S^2)$, and let $p \in S^2$ be at the north pole of the sphere. Let $\theta \in [0,\pi]$, and let $R_{\theta}: T_p (S^2) \rightarrow T_p (S^2)$ be a map that rotates $v$ by $\theta$. Let $w=R_{\theta}(v)$.
We wish to show that such a rotation can be realized a parallel transport along some closed, piecewise regular parametrized curve $\alpha$. We construct $\alpha$ as follows:
1) Move along the meridian defined by $\dot{\alpha}(0) = v$ from p at the north pole to the equator.
2) Move along the equator through the angle $\theta$
3) Move along a meridian back to $p$
We have shown in exercise 4.15 a) that the parallel transport of an arbitrary $v \in T_p (S^2) $ along $\alpha$ makes an angle $\theta$ with respect to $v$, that is, $P_{\alpha} (v) = w $
Since $\alpha$ is composed of three geodesics, it is a piecewise regular parametrized curve, and since for any $\theta \in [0, \pi]$, we have that $P_{\alpha} (v)= R_{\theta} (v) $ for all $v \in T_p ( S^2) $, we thus conclude that $P_{\alpha} = R_{\theta} $, as desired.
$\textbf{My Tutors response}$: "The problem is that your transformation depends on the given $v \in T_p (S^2)$. You need to consider that when your transformation rotates this $v$ through an angle $\theta$, it also rotates all other elements of $T_p (S^2) $ through the same angle."
I do not understand the problem with my transformation. I have chosen an arbitrary element $v \in T_p (S^2)$, and I have shown that a rotation in $T_p (S^2 ) $ of such an element through an arbitrary angle $\theta$ can be realized as the parallel transport of a curve $\alpha$ as constructed above. Why is this not enough? What am I missing?
Thanks in advance