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We show that Ricci flows of invariant metrics on twisted principal bundles with nilpotent fibres over \(S^1\) are immortal and type III. After normalisation and lift to the universal cover, a sequence of metrics converges in the Cheeger--Gromov topology to a homogeneous expanding Ricci soliton. The proof extends the bracket flow technique from homogeneous Ricci flows to invariant Ricci flows on twisted principal bundles.
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Henry Wood (2026) studied this question.
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