I am studying parallel transport in order to understand Berry curvature, but I know this topic is most commonly used in GR so I'm posting my question here. I do not know differential geometry. I am looking for a general explanation of what it means to parallel transport a vector. Mostly I am confused as to what is being held constant as you parallel transport a vector. I initially thought that the components of your transported vector were to be held constant with respect to a Cartesian coordinate system held at the surface of the curve. This coordinate system would always have the x-y axis forming a tangent plane on the surface, and the z axis always directed normal to the surface. But this couldn't be right, because the tangent plane is independent of path as long as the surface is smooth and differentiable. Then in this case, we would never get a different vector direction if we transported our vector around a closed path, which is obviously not true. In other words, I would like to know what is wrong with my reasoning here: Parallel transport of a vector holds the components of the vector constant with respect to a plane tangent to the surface of the curved surface (and also to the axis normal to that surface). The plane tangent to a curved surface at a point will be the same regardless of the path you took to get to that point. Therefore any vector fixed with respect to this plane will also be the same. (which is incorrect) Thank you for your help.