*7.1. Torsion Form*

**Definition 19** (Solder form/soldering of a *G*-principal bundle)**.** *Let π* : *P* → *M be a smooth G-principal bundle over a differentiable manifold M, ρ* : *G* → Aut(*V*) *be a representation, and G be a Lie group.*

*We define the solder form, or soldering, as the vector-valued* <sup>1</sup>*-form <sup>θ</sup>* <sup>∈</sup> <sup>Ω</sup><sup>1</sup> *G* (*P*, *<sup>V</sup>*) *such that* ˜*<sup>θ</sup>* : *TM* <sup>→</sup> *<sup>P</sup>* <sup>×</sup>*<sup>ρ</sup> <sup>V</sup> is a bundle isomorphism, where* ˜*<sup>θ</sup>* <sup>∈</sup> <sup>Ω</sup><sup>1</sup> (*M*, *P* ×*<sup>ρ</sup> V*) *is the associated bundle map induced by the isomorphism of* Ω1 *G* (*P*, *V*) ∼= Ω<sup>1</sup> (*M*, *P* ×*<sup>ρ</sup> V*)*.*
