Let $n$ be an integer with $n\geq 3$, and let $X$ be $S^2$ with $n$ points removed (as a smooth 2-manifold). I am wondering about the following question:
Does there exist a geodesically complete Lorentzian metric on $X$?
We note the answer is already "yes" under any of the following changes to the above requirements:
- if we drop the "geodesically complete" requirement. Then we can embed $X$ as a cylinder with $n-2$ points removed, in Minkowski space $\mathbb{R}^{1|2}$, such that the vertical axis of the cylinder is along the "time" axis; then the pullback of the standard Minkowski metric, to $X$, gives a Lorentzian metric on $X$, but it is not complete due to the $n-2$ "holes".
- if we change "Lorentzian" to "Riemannian". It is well-known that any smooth manifold (without boundary) admits a complete Riemannian metric.
- if $n\in\{1,2\}$ instead of $n\geq3$. In the case $n=2$, $X$ is diffeomorphic to an open cylinder, which embeds as an infinitely long cylinder in $\mathbb{R}^{1|2}$ with cylinder axis along the "time axis", thus getting an induced complete Lorentzian metric by pullback. For the case $n=1$, $X$ is diffeomorphic to just $\mathbb{R}^2$, which already admits a complete Lorentzian metric as the Minkowski space $\mathbb{R}^{1|1}$.
My guess is that the answer to the above question is "no", but I'm not sure how to approach proving this; would anyone have a hint or reference suggestion?
Note: here are the definitions I'm using:
By a "geodesic curve" in a pseudo-Riemannian manifold $M$ (or any smooth manifold with affine connection), I mean a smooth curve $\gamma$ in $M$, with $\mathrm{domain}(\gamma)$ an open interval, such that $\nabla_{\dot{\gamma}} \dot{\gamma} = 0$ everywhere along $\gamma$.
I call a pseudo-Riemannian manifold $M$ "geodesically complete" if any geodesic curve in $M$ can be extended to a geodesic curve whose domain is all real numbers.