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…)