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

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 $E$.

• If you have an $O(k)$ structure then you have a metric on the bundle. For the connection to satisfy $A^T=-A$ it has to be compatible with the metric, i.e., the metric is covariant constant with respect to that connection. – Liviu Nicolaescu May 14 '15 at 23:20

## 2 Answers

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).

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