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For a given smooth bundle, a global section may or may not be available. As I understood smooth vector field (not a local smooth vector field) is a global section of the tangent bundle.

How do we know such a global section exists for a given tangent bundle?

Also, if we have a smooth vector field defined does it mean the bundle is trivial?

Thanks

htr
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    You always have the zero section. – Mariano Suárez-Álvarez Oct 19 '22 at 23:46
  • Thanks. Also, could you describe what happens if the smooth manifold is a Lie group? Do Lie groups always have smooth sections, so that the tangent bundle is trivial? – htr Oct 20 '22 at 00:03
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    Indeed, Lie groups are even parallelizable: a Lie group of dimension $n$ has $n$ global vector fields that are at each point linearly independent. They are obtained by doing translations in the group — you will find a construction in pretty much every introductory textbook (for example, Lee's) – Mariano Suárez-Álvarez Oct 20 '22 at 00:28
  • Thank you, it helped a lot. – htr Oct 20 '22 at 00:31

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