I’ve seen the derivation of the Riemann curvature tensor via the vector R(u,v)w, which essentially gives the rate of change in w as it is parallel transported around the loop defined by u and v as the area of the loop shrinks to zero. This makes sense when u and v are vector fields, as the loop is defined by the flows of u and v (and their Lie bracket to close the curve).
This dependence on the fact that u and v be vector fields reflects itself in the resulting expression for R(u,v)w as the commutator of covariant derivatives (with the Lie bracket term to close the loop) because that construction uses covariant derivatives, so the vector must exist at many different points in space, and thus be a field.
But then I see that this operator R is tensorial in its arguments, so the same value R(u,v)w can be written by contracting indices with the components of the Riemann tensor. But this way of writing it doesn’t make use of the values of u,v,w at any point other than the one you evaluate it at. So why should the definition of R(u,v)w at a point p need u,v,w to be vector fields, and not just vectors at p?
It seems like we’re requiring u,v,w to be vector fields when we don’t need anything besides their values at p anyway.
Edit: At this point my best bet is apparently to exponentiate a u,v parallelogram in the tangent space. I understand what it means to exponentiate a vector into a geodesic, but I don’t understand exactly what it means to exponentiate a parallelogram in the tangent space...