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Let $\pi_P:P\to M$ be a principal $G$-bundle on a manifold $M$ endowed with a connection $A$ and $\pi_Q:Q\to M$ be a principal $H$-bundle on a manifold $M$. Let $f:P\to Q$ be a morphism of bundles along $\lambda :G\to H$ which means that $$\forall p \in P, \forall g\in G: f(p.g)=f(p)\lambda(g)$$ We can push-forward the connection $A$ to a connection $f_*(A)$ on $Q$. Let $a\in\Omega^1(Q,\mathfrak{h})$ be an $\mathfrak{h}$-valued 1-form on $Q$ then $B=f_*(A)+a$ is also a connection on $Q$. We denote the curvature of $A$ by $F_A$.

can we find the curvature of $B$ in terms of $F_A$, $a$ ,$f$ and $\lambda$?

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  • $\begingroup$ If $a=0$ then $F_B=\lambda\cdot F_A$ where $\lambda\cdot\colon\mathfrak g\to \mathfrak h$ is the induced map on Lie algebras. For $a\neq0$ you would in general need also to know $f_*A.$ $\endgroup$
    – Sebastian
    Commented Jul 25, 2013 at 14:38
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    $\begingroup$ @Sebastian: I think, in the case $a=0$ your answer need to revision,$f^*F_B=\lambda.F_A$.(Foundations of differential geometry, Kobayashi vol.1 Prop.6.1) $\endgroup$
    – Amin
    Commented Jul 27, 2013 at 13:54

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