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. Intuitively, soldering expresses in abstract terms the idea that a manifold may have a point of contact with a certain model Klein geometry at each point. In extrinsic differential geometry, the soldering is simply expressed by the tangency of the model space to the manifold. In intrinsic geometry, other techniques are needed to express it. Soldering was introduced in this general form by Charles Ehresmann in 1950.
Soldering of a fibre bundle.
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 form θ is called the solder form for the soldering.
Special cases.
By convention, whenever the choice of soldering is unique or canonically determined, the solder form is called the canonical form, or the tautological form.
Affine bundles and vector bundles.
Suppose that "E" is an affine vector bundle (a vector bundle without a choice of zero section). Then a soldering on "E" specifies first a "distinguished section": that is, a choice of zero section "o", so that "E" may be identified as a vector bundle. 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 Vo"E" ≈ "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. In this case, the structure group of the vector bundle is often implicitly enlarged by the semidirect product of "GL"("n") with the typical fibre of "E" (which is a representation of "GL"("n")).
Principal bundles.
In the language of principal bundles, a solder form on a smooth principal "G"-bundle "P" over a smooth manifold "M" is a horizontal and "G"-equivariant differential 1-form on "P" with values in a linear representation "V" of "G" such that the associated bundle map from the tangent bundle "TM" to the associated bundle "P"×"G" "V" is a bundle isomorphism. (In particular, "V" and "M" must have the same dimension.)
A motivating example of a solder form is the tautological or fundamental form on the frame bundle of a manifold.
The reason for the name is that a solder form solders (or attaches) the abstract principal bundle to the manifold "M" by identifying an associated bundle with the tangent bundle. Solder forms provide a method for studying "G"-structures and are important in the theory of Cartan connections. The terminology and approach is particularly popular in the physics literature.