I need to see if the affirmation is true or false:
If $M$ is a smooth manifold and $p\in M \Rightarrow M-\{p\}$ admits a non complete global smooth field.
Non complete global smooth field.
differential-geometryVector Fields
differential-geometryVector Fields
I need to see if the affirmation is true or false:
If $M$ is a smooth manifold and $p\in M \Rightarrow M-\{p\}$ admits a non complete global smooth field.
Best Answer
This is true. First note that this is true when $M=B(0,r)$ is an open ball in $\Bbb R^n$ and $p=0$. For example consider the unit radial vector field on $M-p$ defined by $$X(a)=\frac{-a}{\Vert a\Vert}.$$ This vector field is not complete, because trajectories will reach $0$ in a finite time.
Now for the general case take a chart $\phi:(B(0,r),0)\longrightarrow (U,p)$ around $p$. You can find a smooth function $\psi :M\to \Bbb R$ which is equal to 1 near $p$ and which is zero outside $V:=M-\phi(\bar{B}(0,r/2))$. If $Y$ is the pushforward of $X$ by $\phi$, which is defined on $U-p$, then the extension of $\psi\cdot Y$ to $M-p$ defines a smooth vector field on $M-p$, which is still incomplete because the trajectories near $p$ will reach $p$ in finite time.