Vector-valued forms inside the first jet bundle

Question

On page 433 of "Self-duality in four-dimensional Riemannian geometry" by Atiyah, Hitchin and Singer, it is written that $p^*(E \otimes \Lambda^1) \subset p^*J_1(E)$, where $\Lambda^1 \to X$ is the bundle of $1$-forms, $p : E^* \to X$ is a vector bundle and $J_1(E) \to X$ is the first jet bundle of $E$. How exactly is this inclusion realized?

Answer 1

An element of $J_1(E)$ can be thought of simply as the value and first derivative of a section of $E$ at a single point. When $E$ is a trivial bundle $M \times V$ so that sections of $E$ are just vector-valued functions $M \to V,$ this is made very explicit by the canonical isomorphism \begin{eqnarray}J_1(E) &=& E \oplus (E \otimes \Lambda^1) \\j^1_x s &\mapsto&(s(x), ds(x)). \end{eqnarray}

For more general vector bundles, to make this same explicit identification we need a way of differentiating sections. If we fix a linear connection $\nabla$ on $E$ then we can simply use $j_x^1s \mapsto (s(x),\nabla s(x)).$ You can check that this is an isomorphism by using local coordinates, where jets are simply Taylor polynomials.

In general, this isomorphism clearly depends on our choice of connection. When restricted to the jets with target zero (i.e. $j^1_xs$ for sections with $s(x)=0$), however, it does not: in local coordinates we have $(\nabla s)_i^\alpha = \partial_i s^\alpha + \omega_{\beta i}^\alpha s^\beta$ where $\omega_{\beta i}^\alpha$ are the connection coefficients of $\nabla;$ so at a point where $s=0$ the formula is simply $(\nabla s)^\alpha_i = \partial_i s^\alpha.$

Thus we have a canonical inclusion $E \otimes \Lambda^1\subset J_1(E)$ given by restricting the isomorphism discussed above to $\{0\} \times (E \otimes \Lambda^1).$ Composing with $p$ yields an inclusion $$p^* (E \otimes \Lambda^1) \subset p^*J_1(E).$$ I haven't checked that this agrees with the formulae given in the paper - it's possible that there's some other way of doing this that I've missed. To check that this really is what the authors mean, I recommend that you carefully verify the equation immediately following the claim of the inclusion.