Time to continue the story for covariant derivatives and parallelism, and do what I promised yesterday on tensors.

Fix a smooth manifold {M} with a connection {\nabla}. Then parallel translation along a curve {c} beginning at {p} and ending at {q} leads to an isomorphism {\tau_{pq}: T_p(M) \rightarrow T_q(M)}, which depends smoothly on {p,q}. For any {r,s}, we get isomorphisms {\tau^{r,s}_{pq} :T_p(M)^{\otimes r} \otimes T_p(M)^{\vee \otimes s} \rightarrow T_q(M)^{\otimes r} \otimes T_q(M)^{\vee \otimes s} } depending smoothly on {p,q}. (Of course, given an isomorphism {f: M \rightarrow N} of vector spaces, there is an isomorphism {M^* \rightarrow N^*} sending {g \rightarrow g \circ f^{-1}}—the important thing is the inverse.) (more…)