Question

In the Riemann curvature tensor, we are supposed to parallel transport the vectors each step of the way. So when we take the first covariant derivative, won’t that be zero? So aren’t we then taking the second covariant derivative of the zero vector?

Answer 1

In calculating the Riemann curvature tensor, the first covariant derivative of a vector field is generally not zero, and the operation in question measures the failure of parallel transport around an infinitesimal loop, rather than taking a second derivative of a zero vector.

Step-by-step explanation:

The question relates to the Riemann curvature tensor in the context of differential geometry, which is a tool used in mathematics and physics to describe the intrinsic curvature of a manifold. When taking the first covariant derivative of a vector field in the process of computing the Riemann curvature tensor, the result is generally not zero unless the vector field is covariantly constant along the direction of differentiation. The operation we are interested in for the Riemann curvature tensor involves the second covariant derivative but applied in a specific manner that measures how much the parallel transport around an infinitesimal loop fails to preserve the vector. This involves comparing the vector field at a point with a parallel transported version of that vector field around an infinitesimal loop, which leads to a measure of curvature and does not generally involve the second derivative of a zero vector.