Principal Bundles by Stephen Bruce Sontz

Author:Stephen Bruce Sontz
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

by the definition of Par t γ . But then is a curve in P covering

Putting t = 0 shows that starts at

Finally,

is horizontal for all t ∈ [0, 1] since is horizontal and the horizontal subspaces are invariant under the right action. In short, satisfies the three properties that uniquely characterize . So we then see that . Equivalently, in the notation of the parallel transport map, we have that

Since this is true for all p 0, we conclude that R g Par t γ  = Par t γ R g , which exactly says that Par t γ is a G-equivariant map. ■

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