Given a smooth $G$-principal bundle $P \to M$, is $P$ in general parallelizable as a manifold? That is, is the tangent bundle $TP \to P$ trivial? In the case of Klein geometries, and more generally, Cartan geometries, the principal bundle is indeed parallelizable, but is this true for any principal bundle?
2025-01-13 05:17:49.1736745469
Is the total space of a principal bundle parallelizable?
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No. $P$ could be the trivial bundle $G \times M$. This won't be parallelizable if $M$ has nonvanishing Stiefel-Whitney or rational Pontryagin classes, by the Kunneth formula.