What significant differences are there between a Riemannian manifold and a pseudo-Riemannian manifold?

Question

I am reading John Lee's book Riemannian Manifolds. On page 91, he begins a chapter called "Geodesics and Distance," which is I think the first chapter that seriously addresses geodesics.

I was very surprised when I came across the following sentence:

Most of the results of this chapter do not apply to pseudo-Riemmanian metrics, at least not without substantial modification.

I thought the only real difference between the two was about the positive-definite constraint. But this makes it sound like there's a whole host of properties that don't apply to pseudo-Riemannian metrics but that do apply to Riemannian metrics.

Can someone clarify this for me? Are the things we can do on Riemannian manifolds that can't be done on pseudo-Riemannian ones?

Answer 1

For one thing, a Lorentz-signature metric on a compact manifold can fail to be geodesically complete. If memory serves, Chapter 3 of Einstein Manifolds by Besse contains an example of a metric on a torus where a finite-length geodesic "winds" infinitely many times.

Generally, the "unit sphere" in a tangent space is non-compact for a metric of indefinite signature (e.g., it's a hyperbola on a Lorentz-signature surface), which can cause all manner of fun.

Answer 2

OK, I'll accept the challenge...

The biggest difference in the pseudo-Riemannian case is that curves can have zero length, and the "Riemannian distance function" (the supremum of the lengths of curves between two points) is not a metric in the sense of metric spaces. Thus most of the results in Chapter 6 of my book don't make sense if the metric isn't positive definite.

Nonetheless, there is still a lot that can be said, at least in the Lorentz case (pseudo-Riemannian metrics of index $1$). In that case, one has to distinguish curves according to the nature of their velocity vectors: a curve $\gamma$ is spacelike if $g(\dot\gamma,\dot\gamma)>0$, lightlike if $g(\dot\gamma,\dot\gamma)\equiv 0$, and timelike if $g(\dot\gamma,\dot\gamma)<0$. The resulting geometric properties, as you might imagine, are intimately connected with the physics of space-time.

But this is a subject that is quite different from Riemannian geometry, which is why I didn't treat it in my book.