I'm studying about vector fields from Schuller's lectures on youtube. Given a smooth manifold $M$, if we have a smooth module $\Gamma(TM)$ consisting of all smooth vector fields on the manifold, it's clear to me that we can't get global "basis vector fields". In other words we can't come up with a global component representation of vector fields (in terms of the basis vector fields) on the entire manifold.
However, I'm unsure if it's possible to always come up with even local basis vector fields.
Given a vector field $X\in\Gamma(TM)$, is it always possible to come up with a local basis representation of the vector field $X$?
Would be grateful if anyone could clarify!
Best Answer
Yes, any chart $(U,x)$ on the manifold $M$ gives rise to a local basis of vector fields defined on $U$,$\left\{\frac{\partial}{\partial x^1}, \dots, \frac{\partial}{\partial x^{m}}\right\}$, where $m= \dim M$; i.e for every $p\in U$, $\left\{\frac{\partial}{\partial x^1}(p), \dots, \frac{\partial}{\partial x^{m}}(p)\right\}$ is a basis for $T_pM$.