Editing Frame bundle (section)

===Smooth frames===
[[Section (fiber bundle)|Local section]]s of the frame bundle of ''<math>M</math>'' are called [[smooth frame]]s on ''<math>M</math>''. The cross-section theorem for principal bundles states that the frame bundle is trivial over any open set in ''<math>U</math>'' in ''<math>M</math>'' which admits a smooth frame. Given a smooth frame ''<math>s: U \to FU</math>'', the trivialization ''<math>\psi: FU \to U \times \mathrm{GL}(n,\mathbb{R})</math>'' is given by
:<math>\psi(p) = (x, s(x)^{-1}\circ p)</math>
where ''<math>p</math>'' is a frame at ''<math>x</math>''. It follows that a manifold is [[Parallelizable manifold|parallelizable]] if and only if the frame bundle of ''<math>M</math>'' admits a global section.

Since the tangent bundle of ''<math>M</math>'' is trivializable over coordinate neighborhoods of ''<math>M</math>'' so is the frame bundle. In fact, given any coordinate neighborhood ''<math>U</math>'' with coordinates ''<math>(x^1,\ldots,x^n)</math>'' the coordinate vector fields
:<math>\left(\frac{\partial}{\partial x^1},\ldots,\frac{\partial}{\partial x^n}\right)</math>
define a smooth frame on ''<math>U</math>''. One of the advantages of working with frame bundles is that they allow one to work with frames other than coordinates frames; one can choose a frame adapted to the problem at hand. This is sometimes called the [[method of moving frames]].