Since this question received no answer, let me go through a simpler case first. Let $P$ be a $U(1)$-bundle and suppose I have a metric on it that makes the fundamental vector field of the action parallel (for example if the metric is invariant under the circle action and the total manifold is Ricci flat). Can I conclude the circle bundle is trivial?
2026-03-28 14:36:59.1774708619
Parallel fundamenal vector fields on circle bundle
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