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Connection on a principal bundle

Contents

Definition

Setup

Let M be a differential manifold, G a Lie group acting on M, and \pi: P \to M a principal G-bundle.

Definition part

A principal G-connection on this principal G-bundle 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 \omega of \Omega^1(P,\mathfrak{g}) such that:

  • Ad(g)(R_g^*\omega) = \omega where R_g denotes right multiplication by g. This condition is G-equivariance
  • If \xi \in \mathfrak{g} and X_\xi is the fundamental vector field corresponding to \xi, then \omega(X_\xi) = xi identically on P.

Related notions

Facts

Viewing a connection on a vector bundle as a principal connection

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Transport using principal connections

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