I am trying to understand covariant derivative and parallel transport.
Suppose the 2D plane with polar coordinates. At a point in the $x$ axis the radial versor $hat{r}$ is horizontal. At a point in the $y$ this versor is vertical. If I parallel transport a vector from the first point to the second, it remains the same. The local versor rotates $\pi/2$ counterclokwise and my vetor rotates $\pi/2$ clockwise with respect to $\hat{r}$.
To me this is the essence of parallel transport. The vector must counter-rotate with respect to the local frame to remain parallel.
However, consider parallel transport of a initially tangent vector to a line of latitude in the sphere, as in the nice animation here. The blue vector is the one being transported. For some latitude in the lower hemisphere, it rotates $\pi/2$ counterclokwise.
Does this mean that the vector which is always tangent to the line of latitude (black) is rotating $\pi/2$ clockwise, when seen from the ambient space? This cannot be, since it returns to itself. How can the rotation of those two vectors compensate?
I can see that the black vector indeed rotates $\pi/2$ clockwise if we take the tangent cone and unfold it. But I can't see this in the sphere itself. How am I supposed to see this in general, when I don't have a cone around to help me?