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A sequence of vector fields on an open subset of a manifold is linearly independent in module sense, i.e., if we multiply by smooth functions. Does it imply the pointwise independence of the vector fields?

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  • $\begingroup$ I'd say not in general. First they should be parallelizable in your manifold, $M$, and then your manifold should admint $n=dim(M)$ parallelizable vectors. $\endgroup$
    – Dox
    Commented Aug 11, 2012 at 12:11

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No: consider the vector fields $\frac{\partial}{\partial x}$ and $x\frac{\partial}{\partial y}$ on $\mathbb R^2$.

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If you add the condition that the submodule generated by those elements is a direct summand then this is true. Proof: the other summand must be projective, so locally free, so at each point the sequence forms part of a basis, so is linearly independent.

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