In mathematics, more precisely in differential geometry, a soldering 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.
Let M be a smooth manifold, and G a Lie group, and letE be a smooth fibre bundle over M with structure groupG. Suppose that Gacts 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 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. Then a soldering on E specifies first a distinguished section: that is, a choice of zero sectiono, 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 VoE ≈ E. Making this identification, the solder form is specified by a linear isomorphism In other words, a soldering on an affine bundleE 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 with the typical fibre of E.
Examples
As a special case, for instance, the tangent bundle itself carries a canonical solder form, namely the identity.
In Hamiltonian mechanics, the solder form is known as the tautological one-form, or alternately as the Liouville one-form, the Poincaré one-form, the canonical one-form, or the symplectic potential.
Applications
A solder form on a vector bundle allows one to define the torsion and contorsion tensors of a connection.
Principal bundles
In the language of principal bundles, a solder form on a smooth principalG-bundle P over a smooth manifold M is a horizontal and G-equivariant differential 1-form on P with values in a linear representationV of G such that the associated bundle map from the tangent bundle TM to the associated bundleP×GV is a bundle isomorphism. 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 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.
