Pullback connection

Let $f:N\to M$ be smooth and let $\mathrm{\nabla }$ be a connection on $E\to M$. Give $f^{\ast }E=\{(q,e)\in N\times E:f(q)=\pi (e)\}$ the subspace topology. If $\mathrm{\Phi }:E{\mathrm{\mid }}_U\to U\times {\mathbb{R}}^r$ is a vector-bundle chart and $\mathrm{\Phi }(e)=(\pi (e),v)$, then $\tilde{\mathrm{\Phi }}(q,e)=(q,v),{\tilde{\mathrm{\Phi }}}^{-1}(q,v)=(q,{\mathrm{\Phi }}^{-1}(f(q),v))\mathrm{.}$ Both displayed maps are continuous in the subspace and product topologies, so this is a homeomorphism from $(f^{\ast }E){\mathrm{\mid }}_{f^{-1}U}$ to $f^{-1}U\times {\mathbb{R}}^r$. On overlaps its change of coordinates is $(q,v)⟼(q,g_{\beta \alpha }(f(q))v),$ which is smooth and fibrewise linear. These charts therefore supply the smooth rank-$r$ bundle structure directly. Its total space is Hausdorff and second countable: $N\times E$ is a smooth manifold by Products of smooth manifolds have a canonical product smooth structure, and both properties pass to the subspace $f^{\ast }E$ by $T_0$, $T_1$, and Hausdorffness are hereditary and Second countability is hereditary.

In a pulled-back frame $f^{\ast }e$ on $f^{-1}U$, the pullback connection $f^{\ast }\mathrm{\nabla }$ is specified by $(f^{\ast }\mathrm{\nabla }{)}_X((f^{\ast }e)u)=(f^{\ast }e)(X(u)+(f^{\ast }\omega )(X)u),$ where $(f^{\ast }\omega {)}_q(v)={\omega }_{f(q)}(d{f}_{q}v)$ entrywise and $u$ is any smooth coefficient column on $N$. These are local prescriptions under the gluing criterion Local connection forms glue exactly when they obey the transformation law; their compatibility and hence well-definedness are proved in the next theorem.

In particular, coefficient functions are not required to factor through $f$. For constant $f$, a constant frame of $E_{f(N)}$ gives zero pulled-back matrix and ordinary differentiation of arbitrary $u$; it does not make every varying section parallel. Empty source and rank-zero bundles use empty coefficient data. No injectivity, immersion, submersion, or choice of an extension is part of this definition.