# Are these equivalent characterizations of closed manifolds?

Let $M$ be a connected smooth manifold without boundary. Are the following equivalent?

1. $M$ is compact
2. $M$ cannot be realized as a proper open subset $M\subset N$ of another connected manifold $N$. In other words, $M$ cannot be made bigger without adding connected components.
3. The flow of every vector field on $M$ is complete.

I recall wondering about this as a student. Some of the implications are well know but I never figured out the others. This question reminded of this.

• Make a infinite chain of tori (gluing with connected sums) so get an infinite chain. This is a non-compact surface: can you embed this as a proper open subset of a connected surface? Sep 24, 2012 at 0:42
• @Mariano: for that surface, yes you can. One way to see this is that this surface is diffeomorphic to a sub-surface, the one you get by removing a properly-embedded half-open interval $[0,\infty)$ from the surface, so $\infty$ goes to the end of the surface. The diffeomorphism is obtained by pushing the end into the surface, a type of "finger move". Sep 24, 2012 at 6:16
• @RyanBudney, cool! Sep 24, 2012 at 7:47
• @Ryan: Can you explain in more detail the "pushing the end into the surface" part of that? I've played around with this answer on other noncompact surfaces and could see how it works, but I'm just not seeing it in this example. Sep 24, 2012 at 20:01
• somewhat similar: math.stackexchange.com/questions/53021/…
– user13618
Sep 24, 2012 at 22:11

3 implies 1 is a little more work but it's a fairly standard argument as well. The idea is that if the manifold is non-compact you can take a proper Morse function on the manifold, $f : M \to [0,\infty)$. If you re-scale the gradient like $\frac{f^2}{1+|\nabla f|^2}\nabla f$, its flow lines can't be complete since it hits infinity in a finite time. And technically you don't need any Morse theory to make this conclusion. It's enough to have a proper function $f : M \to [0,\infty)$ then show you can embed an arc $p : [0,\infty) \to M$ in $M$ so that $f(p(x)) = x$ for all $x$. You can define an incomplete vector field whose flow line is the image of $p$, then extend it to $M$.
• I don't understand the gradient scaling. If the function is $f(x)=x^2$ on $M=\mathbb R$, the flow lines of $\dot x = 2x/(1+x^4)$ are complete.