What is the space for the coefficients of the connection 1-form of a connection in a vector bundle?

Question

Let $E\to X$ is a a (smooth real) vector bundle with structure group some Lie group $G$. Suppose we have a (linear) connection $\nabla$ on $E$.

Is it true that if $A$ is the connection 1-form of the connection $\nabla$ in local coordinates in some trivialisation, regarded as a 1-form with matrix coefficients, then this matrix is in $\mathfrak{g}$ the Lie Algebra of $G$?

When one defines a connection in a principal $G$-bundle one constructs the connection in this way. So it would make sense that if $E$ has structure group $G$ , then this happens too.

However, I have never seen this argument when working with vector bundles only. Furthermore, I have seen proofs trying to deduce properties of the connection $1$-form that would be trivial if we had that the matrices are in the Lie Algebra. So that makes me think that this is not true.

The application I have in mind is the following: If I have a real vector bundle of rank $k$ with structure group the orthogonal group $O(k)$ and $A$ is the connection $1$-form in local coordinates in some trivialisation. Would it always be the case that $A^\top=-A$?

Note: By structure group, I mean the sections of automorphisms defined on the fibres of $E$ that are induced by the transition functions between trivialising neighbourhoods of $E$.

Answer 1

This answer just extends the remark by Liviu Nicolaescu above. For a general connection on a vector bundle with structure group $G$, you cannot say anything about the connection coefficients. (As an extreme example look at the case of a trivial bundle which has structure group $\{e\}$ and still admits lots of non-trivial connections.) But the $G$-structure allows you to define a subclass of linear connections, sometimes called $G$-connections. This definition just requires that in local charts \textit{coming from the $G$-structure} the connection coefficients have values in $\mathfrak g$, which is a well defined condition since it has the same meaning in all charts of the $G$-structure.

However, this condition is equivalent to the linear connection being induced from a principal connection on the $G$-frame bundle of the vector bundle (at least if the vector bundle is induced by an effektive representation of $G$). Hence there is not much gain in working in the vector bundle setting.

For various choices of $G$, the condition on the connection coefficients can be made explicit. For example if $G$ is the stabilizer of some tensor (e.g. $G=O(n)\subset GL(n)$), then this tensor induces a corresponding section of some tensor bundle constructed from $E$ (in the $O(n)$-case a bundle metric, which is a section of $S^2E^*$) and $G$-connections can be characterized as those for which that section is parallel (for the induced connection).

Answer 2

For explicit descriptions of the relation between principal connections and induced connections on associated bundles including vector bundles, see section 19 of this book.