G-structure

Remarks

Example: a Riemannian metric $g$ gives an $\mathrm{O}(n)$-structure, the bundle of $g$-orthonormal frames. Two orthonormal frames at $p$ differ by an orthogonal matrix, so $\mathrm{O}(n)$ acts simply transitively on each fibre. Conversely every $\mathrm{O}(n)$-structure is the orthonormal frame bundle of exactly one Riemannian metric. A $\set{e}$-structure is a global frame, that is, a parallelization of $M$. Non-example: the sphere $S^2$ has no $\set{e}$-structure, since a global frame would contain a nowhere vanishing vector field, which $S^2$ does not have. Other standard cases: $\mathrm{GL}^+(n, \R)$ gives orientations, $\mathrm{GL}(m, \C) \subset \mathrm{GL}(2m, \R)$ gives almost complex structures, and $\mathrm{Sp}(2m, \R)$ gives nondegenerate $2$-forms. A $G$-structure is called integrable, or flat, when near each point there is a chart whose coordinate frames lie in $P$.