----
> [!definition] Definition. ([[structure group of a real vector bundle over a smooth manifold]])
> Let $E$ be a rank-$k$ real [[vector bundle]] over a [[smooth manifold]] $B$. Let $G \subset \text{GL}_{k}(\mathbb{R})$ a [[subgroup]] of the [[general linear group]]. Suppose that there exists a choice of local trivializations $\{ (U_{\alpha}, \Phi_{\alpha}): B=\bigcup_{\alpha}U_{\alpha} \text{ and } \psi_{\beta \alpha}(b) \in G \text{ for all }b \in U_{\alpha} \cap U_{\beta} \}.$
We call such a choice a **choice of a $G$-structure on $E$** and say **$E$ admits a structure group $G$**.
>
The terminology "structure group on a [[smooth manifold]] $M
quot; means a $G$-structure on the [[tangent bundle]] $TM$.
^definition
> [!basicexample]
> 1. If the structure group $G$ is trivial, $G=\{ \id\}$, then all transition functions are identities and so $E$ is a [[product bundle]] (i.e. trivial bundle).
> 2. Having $G=\text{GL}_{+}(k, \mathbb{R})$, the [[subgroup]] of $\det>0$ matrices, gives an [[orientation of a vector bundle|orientation of]] $E$: [[determinant criterion for orientability of a vector bundle]]. Putting $E=TM$ recovers the usual definition of [[orientation of a smooth manifold]].
> 3. If $G=\text{O}(k)$ is the [[general orthogonal group|orthogonal group]], then the standard inner product on $V=\mathbb{R}^{k}$ transfers to an invariantly-defined [[inner product]] on the fibers $E_{p}$ of $E$ (because any change of trivialization is an isometry, and isometries preserve inner products). In this case we call the local trivializations of $G$ **orthogonal trivializations.** Example Sheet 3 shows that any real [[vector bundle]] admits an orthogonal structure.
> 4. Combining the above two: $G=\text{SO}(k)=\text{O}(k) \cap \text{GL}_{+}(k, \mathbb{R})$ amounts to choosing an inner product + orientation.
> 5. Complex and almost complex structures arise too from this construction.
^basic-example
----
####
----
#### References
> [!backlink]
> ```dataview
> TABLE rows.file.link as "Further Reading"
> FROM [[]]
> FLATTEN file.tags as Tag
> WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom"
> GROUP BY Tag
> ```
> [!frontlink]
> ```dataview
> TABLE rows.file.link as "Further Reading"
> FROM outgoing([[]])
> FLATTEN file.tags as Tag
> WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom"
> GROUP BY Tag
> ```