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)). Examples As a special case, for instance, the tangent bundle itself carries a canonical solder form, namely the identity. If M has a Riemannian metric (or pseudo-Riemannian metric), then the covariant metric tensor gives an isomorphism from the tangent bundle to the cotangent bundle, which is a solder form. 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. Solder forms occur in the sigma model, where they glue together the tangent space of a spacetime manifold to the tangent space of the field manifold. Vierbeins, or tetrads in general relativity, look like solder forms, in that they glue together coordinate charts on the spacetime manifold, to the preferred, usually orthonormal basis on the tangent space, where calculations can be considerably simplified. That is, the coordinate charts are the in the definitions above, and the frame field is the vertical bundle . In the sigma model, the vierbeins are explicitly the solder forms. 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