G-structure

Summary

A $G$-structure on an $n$-manifold is a choice, at every point, of a preferred family of frames of the tangent space. Any two preferred frames at a point are related by an element of a fixed subgroup $G$ of $\mathrm{GL}(n, \R)$. Formally it is a reduction of the frame bundle to the group $G$. Many geometric structures are $G$-structures: Riemannian metrics for $G = \mathrm{O}(n)$, orientations for $\mathrm{GL}^+(n, \R)$, almost complex structures for $\mathrm{GL}(m, \C)$ with $n = 2m$. The notion gives one language for comparing and classifying these structures.