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YCor
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Non existence-existence of a parametrically compatible metric to a complete geodesible vector field on $\mathbb{R}^2\setminus\{p,q\}$

Inspired by this answer to the question entitled "Possible isometry groups of open manifolds" we ask the following question:

Is there a complete vector field $X$ on $\mathbb{R}^2\setminus\{p,q\}$ whose foliation is a geodesible foliation but there is no a Riemannian metric $g$ on $\mathbb{R}^2\setminus\{p,q\}$ such that all trajectories of $X$ are length parametrized geodesics?  ($|X|_g=1$)

Non existence of a parametrically compatible metric to a complete geodesible vector field on $\mathbb{R}^2\setminus\{p,q\}$

Inspired by this answer to the question entitled "Possible isometry groups of open manifolds" we ask the following question:

Is there a complete vector field $X$ on $\mathbb{R}^2\setminus\{p,q\}$ whose foliation is a geodesible foliation but there is no a Riemannian metric $g$ on $\mathbb{R}^2\setminus\{p,q\}$ such that all trajectories of $X$ are length parametrized geodesics?($|X|_g=1$)

Non-existence of a parametrically compatible metric to a complete geodesible vector field on $\mathbb{R}^2\setminus\{p,q\}$

Inspired by this answer to the question entitled "Possible isometry groups of open manifolds" we ask the following question:

Is there a complete vector field $X$ on $\mathbb{R}^2\setminus\{p,q\}$ whose foliation is a geodesible foliation but there is no a Riemannian metric $g$ on $\mathbb{R}^2\setminus\{p,q\}$ such that all trajectories of $X$ are length parametrized geodesics?  ($|X|_g=1$)

Post Undeleted by Stefan Kohl♦
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Ali Taghavi
Ali Taghavi
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Non existence of a parametrically compatible metric to a complete geodesible vector field on $\mathbb{R}^2\setminus\{p,q\}$

Inspired by this answer to the question entitled "Possible isometry groups of open manifolds" we ask the following question:

Is there a complete vector field $X$ on $\mathbb{R}^2\setminus\{p,q\}$ whose foliation is a geodesible foliation but there is no a Riemannian metric $g$ on $\mathbb{R}^2\setminus\{p,q\}$ such that all trajectories of $X$ are length parametrized geodesics?($|X|_g=1$)