Tuesday, July 13, 2010

Connection principal bundle

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 \mathfrak g 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 \Omega^1(P,\mathfrak g)\cong C^\infty(P, T^*P)\otimes\mathfrak g such that
  1. \hbox{Ad}(g)(R_g^*\omega)=\omega where Rg denotes right multiplication by g;
  2. if \xi\in \mathfrak g 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.