= Curvature of a pullback connection
{title2=$R^{\phi^*\nabla}=\phi^*R^\nabla$}
For a <smooth map> $\phi:M'\to M$ and a <connection on a vector bundle> $E\to M$, the <curvature form of a connection> on the <pullback vector bundle> satisfies
$$
R^{\phi^*\nabla}_p(u,v)=R^\nabla_{\phi(p)}(d\phi_pu,d\phi_pv)
$$
as an endomorphism of $E_{\phi(p)}$. In a <frame of a vector bundle> written as a row, with section components written as columns, the <connection one-form> is $\Omega$ and its <curvature form of a connection> is $d\Omega+\Omega\wedge\Omega$. The <pullback connection> has matrix $\phi^*\Omega$; commutation of the <pullback of a differential form> with the <exterior derivative> and <wedge product of differential forms> proves the identity. No pushforward to an ambient <vector field> is required: $d\phi(X)$ is a <vector field along a map>.
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