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We know from Riemannian geometry that for Riemannian manifolds, completeness and geodesic completeness are equivalent, which is usually a consequence of Hopf-Rinow theorem. However, I'm considering a more conceptual reformalization of this fact. Let's consider the simpler direction for this.

Suppose $M$ is a Riemannian manifold and $UM$ is its unit sphere bundle. Levi-Civita connection gives an horizontal vector field $W$ on $UM$, which determines the local geodesic flow. If $M$ is complete, we need to show that the geodesic flow on $UM$ is complete, i.e. integral curves are indefinitely extendable.

I guess that it will follow from a more general result on fiber bundles and vector fields on it. For example, we know that the fibers of $UM\to M$ are compact, and $W$ is horizontal. It's just like the theorem on completeness of flows of vector fields on compact manifolds. It seems a more natural way to formalize the statement.

Any help? Thanks!

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    $\begingroup$ This question on the Hamiltonian formulation of geodesics and long time existence on a compact manifold is relevant. $\endgroup$ – aes Dec 31 '14 at 16:16
  • $\begingroup$ What you should keep in mind is that the geodesic flow on compact Lorentzian manifolds is, in general, incomplete. This will pretty much kill any "conceptual" proof of the Hopf Rinow theorem. $\endgroup$ – Moishe Cohen Dec 31 '14 at 21:59
  • $\begingroup$ @studiosus In that case, the fibers aren't compact. That's why I mentioned the unit sphere bundle. $\endgroup$ – Frank Science Jan 1 '15 at 15:09
  • $\begingroup$ @aes Thanks! I don't know anything on symplectic geometry, but in the case that $M$ is compact, $UM$ is compact and everything is easier. I wonder how anything goes when compactness is replaced with completeness. $\endgroup$ – Frank Science Jan 1 '15 at 15:16
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    $\begingroup$ The vector field $V(x) = (1 + x^2) d/dx$ on $\mathbb{R}$ seems to give rise to a horizontal vector field on $U\mathbb{R}$ whose integral curves aren't infinitely extendable (since the integral of $1/(1+x^2)$ converges). I think $V(x)$ as above can even be thought of as a parallel field for a suitably defined connection on $\mathbb{R}$. This example makes it hard for me to see how one might generalize the Hopf-Rinow theorem without at least some notion of length. (There is a generalization of Hopf-Rinow to certain types of length-metric spaces, including e.g. Finsler manifolds.) $\endgroup$ – mollyerin Jan 2 '15 at 3:27
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Just in order to close this matter:

  1. There are two easier existence theorems in differential geometry:

a. If $E\to M$ is a fiber bundle (say, a principal bundle, for concreteness) equipped with a connection $\nabla$, then each smooth curve on $M$ lifts to a smooth horizontal curve in $E$.

b. If $M$ is a compact Finsler manifold and $\nabla$ is a compatible affine connection, then geodesic flow of $\nabla$ exists for all values of the time parameter.

Both theorems are simple application of existence-uniqueness theorems for ODEs on manifolds; in the 2nd case, the relevant theorem is:

Theorem. Let $U$ be a smooth compact manifold and $X$ be a smooth vector field on $U$. Then $X$ gives rise to a 1-dimensional group of diffeomorphisms (a flow) on $U$, $f_t: U\to U$, $\frac{d}{dt}f(u)=X(u)$, $u\in U$.

  1. In contrast, Hopf-Rinow theorem is a much-much harder result, which relates metric geometry of Riemannian (more generally, Finsler) manifold to its geodesic flow. The direction from metric completeness to geodesic completeness is the easier one. The metric completeness assumption is a tool which allows one to essentially reduce the existence of the flow to the compact case (when you are dealing with compact subsets of the unit tangent bundle). In the process of the proof, one has to establish several important things which are not at all obvious (and which make the proof complicated):

a. Relation of distance-minimizing curves to geodesics (that they are the same locally).

b. Relation of the metric topology of $M$ to its manifold topology (that they are the same).

However, even given (a) and (b), the proof still requires some trickery.

  1. As an aside, the notion of a "geodesic" makes sense beyond Finsler geometry, in the context of affine structures (affine connections on the tangent bundle of $M$) and, even more generally, for projective and conformal structures (this is discussed, I think, in Nomizu's book "Transformation groups"). Compactness of the manifold $M$ no longer guarantees geodesic completeness. Nevertheless, the question "to which extent compactness implies geodesic completeness" is important and remains a subject of active research. For instance, here is a little-known theorem of Y.Carriere (1989) who proved

Theorem. Compact flat Lorentzian manifolds are geodesically complete.

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