As far as I know, principal bundle is a fiber bundle with a fiber beeing a principal homogeneous space (or a topological group). According this definition vector bundle is a special principal bundle, because vector space with vector addition as group operation is a topological group.(adsbygoogle = window.adsbygoogle || []).push({});

But I feel that something is wrong with this, because there is a theorem that a principal bundle is trivial if and only if it has a global section (see C. Nash, S. Sen: Topology and Geometry for Physicists, p. 152). Möbius strip (regarded as vector bundle) has a global section (the identically 0 section) but it isn't trivial. Where is the error?

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# Are vector bundles principal bundles?

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