Formal Definition
Let π:P→M be a smooth principal G-bundle over a smooth manifold M. Then a principal G-connection on P is a differential 1-form on P with values in the Lie algebra of G which is G-equivariant and reproduces the Lie algebra generators of the fundamental vector fields on P.
In other words, it is an element ω of such that
- where Rg denotes right multiplication by g;
- if and Xξ is the vector field on P associated to ξ by differentiating the G action on P, then ω(Xξ) = ξ (identically on P).
Sometimes the term principal G-connection refers to the pair (P,ω) and ω itself is called the connection form or connection 1-form of the principal connection.
Read more about this topic: Connection (principal Bundle)
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