I know that locally that is possible, as for local coordinates $x_i$ $(\frac{\partial}{\partial x_i})_{i \in I}$ spans $T_p M$ and therefore, one gets a local vector field. How do I extend this one?
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$\begingroup$ Since you are asking for the possibility of always finding such a "global basis", do you have one example at least where it is possible? $\endgroup$ – uniquesolution Jun 28 '17 at 20:39
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$\begingroup$ I was reading in Bredon, where he just takes a basis $X_0 , \ldots , X_n$ for $T_e G$, where $G$ is a Lie group. $\endgroup$ – MPB94 Jun 28 '17 at 20:44
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2$\begingroup$ Assuming you want smooth vector fields, then no it's not always possible. For Lie groups, since left multiplication is a diffeomorphism, you can take a basis for the tangent space at the identity, and "translate" it to a basis at every point. This defines smooth vector fields that form a basis at every point. $\endgroup$ – Steve D Jun 28 '17 at 20:48
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In general, you are not even guaranteed to find one global vector field which never vanishes. This is the case for $M=S^{2n}$ (hairy ball theorem).
