Why is the parallel transport on the 2-sphere SO(3)-equivariant? I am trying to prove the following equation for $R \in SO(3)$:
\begin{equation*}
R^{-1}P_{R(\gamma)}(R(v)) = P_{\gamma}(v)
\end{equation*}
where $\gamma \colon \lbrack 0,1 \rbrack \longrightarrow S^2$ is a curve and $P_{\gamma}$ the parallel transport along $\gamma$.
I have some kind of proof, but I think it can't be correct, because it only uses the fact that $R \colon S^2 \longrightarrow S^2$ is well defined and that $R$ is a diffeo. Here is a sketch:
I used the fact, that the parallel transport is uniquely determined by the unique parallel vector field $X$ with $X(0) = v$ and $\nabla_t X = 0$ on $\lbrack 0,1 \rbrack$. So I chose a map $(x,U)$ around $p$ and wrote down the differential equations in the basis belonging to the map $(x \circ R^{-1}, R(U))$, which is just given by $R(X_i)$, when $X_i$ are the basis vectors for $T_p S^2$ corresponding to $x$ (is that correct?).
Then I could use linearity of $R$ to show that if I multiply with $R^{-1}$ the differential equations for the parallel transport $P_{\gamma}$ are fullfilled. 
Something must be wrong, because I dont use $\det(R) = 1$ at all. Any hints, where this is needed, would be very helpfull..
Thanks a lot.
 A: The differential equation for parallel transport involves the Riemannian metric.  So your proof will have to use the fact that $R$ preserves the metric (i.e. that if
$x \in S^2$ and $v,w$ are tangent vectors at $x$, then $g_{R(x)}(R_*v, R_*w) 
= g_x(v,w)$).  On the other hand, this is all that will be needed (together with the 
fact that $R$ is smooth, so that it respects the various derivatives and so on that
are involved).
The fact that det $R = 1$ isn't important; a reflection about a plan through the origin will also preserve parallel
transport (since it preserves the metric).  The key thing is that $R$ does in fact preserve the metric on $S^2$, which in turn follows from the fact that $R$ is the
restriction to $S^2$ of an isometry of $\mathbb R^3$.  (You might want to first check that $R$ preserves the Riemannian metric on $\mathbb R^3$, which will follow
essentially by definition.  Then check that if $R$ is an isometry of some ambient manifold that
preserves a submanifold $M$, then it preserves the induced metric on $M$.)
