Editing Frame bundle (section)

==Tangent frame bundle==
The '''tangent frame bundle''' (or simply the '''frame bundle''') of a [[smooth manifold]] ''<math>M</math>'' is the frame bundle associated with the [[tangent bundle]] of ''<math>M</math>''. The frame bundle of ''<math>M</math>'' is often denoted ''<math>FM</math>'' or ''<math>\mathrm{GL}(M)</math>'' rather than ''<math>F(TM)</math>''. In physics, it is sometimes denoted ''<math>LM</math>''. If ''<math>M</math>'' is ''<math>n</math>''-dimensional then the tangent bundle has rank ''<math>n</math>'', so the frame bundle of ''<math>M</math>'' is a principal ''<math>\mathrm{GL}(n,\mathbb{R})</math>'' bundle over ''<math>M</math>''.

===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]].

===Solder form===
The frame bundle of a manifold ''<math>M</math>'' is a special type of principal bundle in the sense that its geometry is fundamentally tied to the geometry of ''<math>M</math>''. This relationship can be expressed by means of a [[vector-valued differential form|vector-valued 1-form]] on ''<math>FM</math>'' called the '''[[solder form]]''' (also known as the '''fundamental''' or [[tautological one-form|'''tautological''' 1-form]]). Let ''<math>x</math>'' be a point of the manifold ''<math>M</math>'' and ''<math>p</math>'' a frame at ''<math>x</math>'', so that
:<math>p : \mathbf{R}^n\to T_xM</math>
is a linear isomorphism of '''''<math>\mathbb{R}^n</math>''''' with the tangent space of ''<math>M</math>'' at ''<math>x</math>''. The solder form of ''<math>FM</math>'' is the '''''<math>\mathbb{R}^n</math>'''''-valued 1-form ''<math>\theta</math>'' defined by
:<math>\theta_p(\xi) = p^{-1}\mathrm d\pi(\xi)</math>
where ξ is a tangent vector to ''<math>FM</math>'' at the point ''<math>(x,p)</math>'', and ''<math>p^{-1}: T_x M \to \mathbb{R}^n</math>'' is the inverse of the frame map, and ''<math>d\pi</math>'' is the [[pushforward (differential)|differential]] of the projection map ''<math>\pi: FM \to M</math>''. The solder form is horizontal in the sense that it vanishes on vectors tangent to the fibers of ''<math>\pi</math>'' and [[equivariant|right equivariant]] in the sense that
:<math>R_g^*\theta = g^{-1}\theta</math>
where ''<math>R_g</math>'' is right translation by ''<math>g \in \mathrm{GL}(n,\mathbb{R})</math>''. A form with these properties is called a basic or [[tensorial form]] on ''<math>FM</math>''. Such forms are in 1-1 correspondence with ''<math>TM</math>''-valued 1-forms on ''<math>M</math>'' which are, in turn, in 1-1 correspondence with smooth [[bundle map]]s ''<math>TM \to TM</math>'' over ''<math>M</math>''. Viewed in this light ''<math>\theta</math>'' is just the [[identity function|identity map]] on ''<math>TM</math>''.

As a naming convention, the term "tautological one-form" is usually reserved for the case where the form has a canonical definition, as it does here, while "solder form" is more appropriate for those cases where the form is not canonically defined. This convention is not being observed here.