I'm reading Donaldson's book, Floer homology groups in Yang-Mills theory. On page 82, he considers a trivial bundle $P$ over a $4$-manifold $X$ with tubular ends which is equipped with a connection $A_0$ and this connection is flat on each ends. (This is called an adapted bundle in the book.) He writes the covariant derivative induced by $A_0$ as $\nabla_0$ and on page 84, he apply this $\nabla_0$ to a gauge transformation $g$. My question is that since he defines the gauge transformation group as Aut($P$), how we can apply $\nabla_0$ to such an element? I only know how to apply $\nabla_0$ to sections of those vector bundles associated to $P$. And I don't know how I can regard elements in Aut($P$) as such sections? In short, what is $\nabla_0 g$ for $g \in$ Aut($P$)?

Thank you.


If I understand correctly, $A_0$ is a connection on a principal bundle $P \to X$, while $\nabla_0$ is the corresponding connection on some associated vector bundle $E \to X$.

The connection $\nabla_0$ induces a connection (denoted, say, $\nabla'$) on the vector bundle $\text{End}(E) \to X$. Meanwhile, I'm imagining that the gauge transformation $g \in \text{Aut}(P)$ induces vector bundle morphism $E \to E$, which gives a morphism $\Gamma(E) \to \Gamma(E)$, and hence an element $g' \in \text{End}(\Gamma(E)) \cong \Gamma(\text{End}(E))$.

So, I'm speculating that $\nabla_0 g$ means $\nabla' g'$.


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