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I would like to have a concrete proof or reference to the following fact:

Let $P\rightarrow M$ be a principal $G$-bundle over an $n$-dimensional manifold $M$, and let $E=P\times_\rho V$ be an associated vector bundle to $P$ and the representation $\rho:G\rightarrow\text{GL}(V)$. Then, a principal connection on $P\rightarrow M$ determines a covariant derivative $$\nabla:\Omega^0(M,E)\rightarrow\Omega^1(M,E)$$ on the associated vector bundle $E$. Moreover, each covariant derivative on $E\rightarrow M$ can be obtained from a unique principal connection on $P\rightarrow M$.

Here by principal connection we understand an Ehresmann connection on $P$ that is $G$-invariant on the decomposition of $TP$. For example, in Mathematical Gauge Theory with Applications to the Standard Model of Particle Physics by Mark J. D. Hamilton, only one direction is dicussed throughout sections 8 and 9 of chapter 5, that a principal connection determines a covariant derivative via the notion of parallel transport. However, there is no further discussion on how to construct a principal connection from a covariant derivative.

This, being a rather classical question, should be treated extensively somewhere. I would like to have a reference of this result, or the half that is not covered in the aforementioned book of Hamilton.

Thanks in advance for your answers.

Akerbeltz
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1 Answers1

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There are two issues here. First, the construction of an induced connection does not work for general Ehresmann connections but only for principal connections, which are compatible with the principal right action on $P$. This may be a matter of terminology only.

More importantly, the converse assertion that you are looking for simply is not true in general. It is true if $P$ is the full linear frame bundle of the vector bundle $E$ so $G=GL(k,\mathbb R)$, where $k$ is the rank of $E$. For example, assume that $(M,g)$ is a Riemannian manifold and $P$ is the orthonormal frame bundle of $M$, so $G=O(n)$ where $n=\dim(M)$ and $P\times_G\mathbb R^n=TM$. Then it is a classical result that a linear connection on $TM$ is induced by a principal connection on $P$ if and only if it is metric for $g$ in the sense that $\xi\cdot g(\eta,\zeta)=g(\nabla_\xi\eta,\zeta)+g(\eta,\nabla_\xi,\zeta)$ for all vector fields $\xi,\eta,\zeta$. The fact that induced linear connections have additional properties is one of the reasons why principal bundles are useful and not just an equivalent encoding of vector bundles.

Andreas Cap
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  • I have edited my question. Apparently, I have been using terms that are not standard at all. – Akerbeltz Feb 03 '22 at 16:53
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    I ought that you actually meant principal connections. But the second issued remains unchanged. Whether the map sending principal connections on $P$ to covariant derivatives on $E$ is injective or surjective depends on what $G$ and the representation of $G$ on $V$ that leads to $E$ are. For the linear frame bundle of $E$ the map is bijective, in general it is neither injective nor surjective. – Andreas Cap Feb 04 '22 at 07:58