Editing Principal bundle (section)

===Characterization of smooth principal bundles===

If <math>\pi : P \to X</math> is a smooth principal <math>G</math>-bundle then <math>G</math> acts freely and [[proper map|properly]] on <math>P</math> so that the orbit space <math>P/G</math> is [[diffeomorphic]] to the base space <math>X</math>. It turns out that these properties completely characterize smooth principal bundles. That is, if <math>P</math> is a smooth manifold, <math>G</math> a Lie group and <math>\mu : P \times G \to P</math> a smooth, free, and proper right action then
*<math>P/G</math> is a smooth manifold,
*the natural projection <math>\pi : P \to P/G</math> is a smooth [[submersion (mathematics)|submersion]], and
*<math>P</math> is a smooth principal <math>G</math>-bundle over <math>P/G</math>.