I came across this question:
"A 2d Riemannian manifold has metric
$$ ds^2=dr^2+f(r)^2d\phi^2 $$ where $\phi$ is identified periodically with period $2\pi$.
Determine the necessary and sufficient conditions on $f(r)$ for the circle $r=r_0$ to have the property that all vectors are invariant under parallel transport."
My attempt simply involves noting that if a vector $Y$ is parallel transported along a curve with vector $X$, then $\nabla_XY=0$.
I get the connection coefficients to be zero except $\Gamma^r_{\phi\phi}=-f^\prime(r)f(r)$, hence to satisfy the above condition, one gets
$$ X^\nu\partial_\nu Y^r- f^\prime (r_0) f(r_0)X^\phi Y^\phi=0$$
Since $X$ is tangent to the circle, then $X^\phi=1$ and zero for all other components.
However, I can't see how $f(r)$ can be chosen to make this is true for all vectors $Y$.
You missed a few of the Christoffel symbols. In particular, $\Gamma^\phi_{r\phi}=\Gamma^\phi_{\phi r} = f'(r)/f(r)$. You should get a constant coefficient second order differential equation for $Y^r$, like $$\partial^2_\phi Y^r + f'(r_0)^2 Y^r = 0,$$ so now you should be able to take it from here.