Solder form

In mathematics, more precisely in differential geometry, a soldering (or sometimes solder form) of a fiber bundle to a smooth manifold is a manner of attaching the fibers to the manifold in such a way that they can be regarded as tangent.

In extrinsic differential geometry, the soldering is simply expressed by the tangency of the model space to the manifold.

[1] Let M be a smooth manifold, and G a Lie group, and let E be a smooth fibre bundle over M with structure group G. Suppose that G acts transitively on the typical fibre F of E, and that dim F = dim M. A soldering of E to M consists of the following data: In particular, this latter condition can be interpreted as saying that θ determines a linear isomorphism from the tangent space of M at x to the (vertical) tangent space of the fibre at the point determined by the distinguished section.

The solder form is then a linear isomorphism However, for a vector bundle there is a canonical isomorphism between the vertical space at the origin and the fibre VoE ≈ E. Making this identification, the solder form is specified by a linear isomorphism In other words, a soldering on an affine bundle E is a choice of isomorphism of E with the tangent bundle of M. Often one speaks of a solder form on a vector bundle, where it is understood a priori that the distinguished section of the soldering is the zero section of the bundle.

Solder forms provide a method for studying G-structures and are important in the theory of Cartan connections.