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

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.
