Suppose you have a Vector field $X$ on a smooth (complete) manifold $M$, whose flow $\phi_X^t$ is, for each time $t\in (-\varepsilon,\varepsilon)$, smooth (of class $C^k$).

Questions: Is $X$ smooth (of class $C^{k-\textrm{something}}$)? What about the case $M=\mathbb{R}^n$?

Thank you in advance!

Suppose you have a Vector field $X$ on a smooth (complete) manifold $M$, whose flow $\phi_X^t$ is, for each time $t\in (-\varepsilon,\varepsilon)$, smooth (of class $C^k$).

Questions: Is $X$ smooth (of class $C^{k-\textrm{something}}$)? Can I control that "something"? What about the case $M=\mathbb{R}^n$?

Thank you in advance!

Suppose you have a Vector field $X$ on a smooth (complete) manifold $M$, whose flow $\phi_X^t$ is, for each time $t\in (-\varepsilon,\varepsilon)$, smooth (of class $C^k$).

Questions: Is $X$ smooth (of class $C^k$)? What about the case $M=\mathbb{R}^n$?

Thank you in advance!

Suppose you have a Vector field $X$ on a smooth (complete) manifold $M$, whose flow $\phi_X^t$ is, for each time $t\in (-\varepsilon,\varepsilon)$, smooth (of class $C^k$).

Questions: Is $X$ smooth (of class $C^{k-\textrm{something}}$)? What about the case $M=\mathbb{R}^n$?

Thank you in advance!