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$ω_1$ and $ω_2$ are two principal connection form on a principal G-bundle $P→M$. $E→M$ is an associated vector bundle to $P$. $∇_1$ and $∇_2$ are the two linear connections on $E$ corresponded to $ω_1$ and $ω_2$.

Then, $ω_1-ω_2$ is a tensorial or basic (means G-equivariatn and horizontal) $\mathfrak g$ valued 1-form on $P$ and so it can be identified with an $adP$ valued 1-form on $M$ where $adP$ is the adjoint bundle of $P$.

Also, $∇_1-∇_2$ can be identified with an $End(E)$ valued 1-form on $M$.

It seems that the above $adP$ valued 1-form on $M$ and $End(E)$ valued 1-form on $M$ are indeed the same thing and there should be a natural identification between them.

How to define the natural identification? Is there also an intuitive explaination?

jw_
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1 Answers1

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They are not the same. When you associate the vector bundle $E$ to $P$ you choose a representation of $G$ on some space $W$, so $W$ ends up being the model fiber for $E$. So choose $W$ to be the zero space and on it the zero representation, then this cannot be all the same as $\text{Ad}(P)$.

What you can instead find, choosing the representation $\rho:G\to\text{GL}(W)$, there is a bundle homomorphism $$\rho_*:\text{Ad}(G)\to\text{End}(E)$$ which acts trivially on the base. I'm not completely sure right now, but I think that if $\rho$ is injective/surcective/an isomorphism, then $\rho_*$ is the same in each fiber. This bundle homomorphism then extends to a homomorphism of $\text{Ad}(P)$-valued to $\text{End}(E)$-valued 1-forms on $M$.

Finally one can note that if $P$ is already the frame bundle of a given bundle $E$ and you associate $E$ to $P$ again via the identity representation $\rho$, then also $\rho^*$ is essentially the identity map. This may have given you the impression of that "there should be a natural identification between them".

nicrot000
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