Let $\pi:P\to M$ be a principal $G$-bundle and $K$ a maximal compact subgroup of $G$. Then, $G/K$ is homeomorphic to a vector space $V$, so $P/K\to M$ is a fibre bundle whose fibres are homeomorphic to $V$. Is there a natural vector bundle structure on $P/K\to M$?
2026-03-25 23:35:33.1774481733
Vector bundle obtained by a maximal compact subgroup
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