In my recent studies of curvature, I worked with the Riemann tensor and the equation: ([itex]\delta[/itex]V)a= A[itex]\mu[/itex]B[itex]\nu[/itex]Rab[itex]\mu[/itex][itex]\nu[/itex]Vb Now previously, I worked in 2D with the 2 sphere. While doing so, I learned that if I set my x1 coordinate to be θ and my x2 coordinate to be ø, then the vectors that serve to be the legs of the loop that I am transporting around would be as follows: A[itex]\mu[/itex] = <θ, 0> B[itex]\nu[/itex] = <0, ø> and then of course the vector that I parallel transport would be as follows: Vb= <θ, ø> Now this may work for 2 dimensions, but what if I have 3 or more dimensions? With only 2 vectors being the legs of the loop, there wouldn't be enough vectors for me to give each individual coordinate its own leg with every other component being 0 (as shown above with A[itex]\mu[/itex] and B[itex]\nu[/itex]). How do I deal with this? Is it even a requirement for every coordinate to have its own leg that is reminiscent of a unit vector? Page 5 on the following PDF gave me the impression that it is a requirement: http://www.physics.ucc.ie/apeer/PY4112/Curvature.pdf Is it possible for one of the legs of the loop to have more than one type of coordinate in it (like A[itex]\mu[/itex] = <r , θ, 0>) ?