Forming a vector field if a tangent vector is given.

Question

Let $M$ be a smooth manifold. Let a tangent vector $v$ be given corresponding to a point $p$ in $M$.

Is there a way to extend it to a whole smooth vector field over $M$? And if so, what is such an EXPLICIT vector field?

Answer 1

EDIT: I misread your question.

Yes, there is a way to extend it. Let $v_p\in T_pM$ and let $(\phi, U)$ be a local coordinate chart containing $p$. You can then construct a smooth bump function $\psi: U\rightarrow \mathbb{R}$ for which $\psi(p)=1$ and the support of $\psi$ is a compact set inside of $U$.

Then write $v$ in terms of the coordinate vector fields of $\phi$ at $p$. $$ v_p = \sum_{i=1}^n v^i(p)\frac{\partial}{\partial x^i}\Big|_{p}. $$

and extend to a vector field on $U$ by defining the coefficient functions on $U$ to be constant $v^i$ $$ V = \sum_{i=1}^n v^i\frac{\partial}{\partial x^i}. $$

Then multiply $V$ by $\psi$ and the result $\psi V$ is a smooth vector field on $U$ for which $V_p = v_p$ and the support of $V$ is strictly contained in $U$.

We then can extend $\psi V$ to the whole manifold by specifying the values outside of $U$ to be zero, since the values outside the support of $\phi V$ in $U$ are zero as well.