I'm currently learning about parallel transport and connections and we were considering the parallel transport of a tangent vector along a sphere as given in the picture below.
From my understanding, by defining a connection on your manifold, you provide a way to identify vectors at one point of the manifold with vectors at another point on the manifold via parallel transporting the vector.
So in the given example, when the initial vector is parallely transported along the closed curve it returns to the same spot as a different vector. Is this because there has not been defined a correct connection on the sphere (that takes into account the curvature of the 2-sphere)? In which case, when a connection is defined on the sphere, (i.e. by the covariant derivative) parallel transport of any vector along a closed curve back to its initial position will result in the same vector?
So this is in fact an example of the need for a connection, and not just the standard derivative?
Thanks in advance!