Pullback connection on a vector bundle

Let $f:N\to M$ be a smooth map, let $E\to M$ be a smooth real or complex vector bundle, and let $\nabla$ be a connection on $E$ . The pullback connection $f^*\nabla$ is the unique connection on the pullback bundle $f^*E\to N$ such that

(f^*\nabla)(f^*s)=f^*(\nabla s)

for every local smooth section $s$ of $E$ . Here the right side is the pullback of the $E$ -valued $1$ -form $\nabla s$ , so it is a $f^*E$ -valued $1$ -form on $N$ .

Explicitly, for an $E$ -valued one-form $\beta$ , its pullback is $(f^*\beta)_y(v)=\beta_{f(y)}(df_yv)$ , regarded as an element of $(f^*E)_y=E_{f(y)}$ .

Local construction

Every local section of $f^*E$ can be written as a finite sum $\sigma=\sum_i a_i f^*s_i$ . The Leibniz rule forces

(f^*\nabla)\sigma = \sum_i da_i\otimes f^*s_i +\sum_i a_i f^*(\nabla s_i).

This formula is independent of the chosen expression for $\sigma$ . In a local frame with $\nabla=d+A$ , the pullback connection is $d+f^*A$ .

Curvature and functoriality

Curvature is natural under pullback:

R^{f^*\nabla}=f^*R^\nabla.

Thus a pullback of a flat connection is flat, although the converse need not hold because $f$ may miss directions on which the original curvature is nonzero. Pullback is functorial: for $g:L\to N$ , the canonical identification $(f\circ g)^*E\cong g^*f^*E$ carries $(f\circ g)^*\nabla$ to $g^*(f^*\nabla)$ . These naturality statements follow from the local pullback formula.

If $i:S\hookrightarrow M$ is an embedded submanifold, $i^*\nabla$ is the restriction of $\nabla$ to directions tangent to $S$ .

Examples and scope

For a constant map $f:N\to\{x\}\subset M$ , the pullback bundle is canonically $N\times E_x$ , and $f^*\nabla$ is the trivial connection in this identification. Pulling back the Levi–Civita connection along a curve gives the covariant derivative used to define parallel vector fields along that curve.

References

Loring W. Tu, Differential Geometry: Connections, Curvature, and Characteristic Classes, Springer, 2017. DOI record. Relevant: §25, pullback connections and curvature.
Shoshichi Kobayashi and Katsumi Nomizu, Foundations of Differential Geometry, Volume I, Wiley Classics, 1996. Publisher record. Relevant: Chapter II, mappings and induced connections.