Suppose I have a principal bundle $P$ relative to the group $G$. Suppose I have a torsionless connection on $TP$ for which the fundamental vector fields relative to the $G$ action are parallel. Can I conclude that the bundle is trivial? If this is true can I conclude the connection is a product connection?
2026-03-28 12:56:01.1774702561
Parallel fundamental vector fields
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