Editing Frame bundle (section)

===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.