In a smooth pseudo-riemannian manifold like those in GR, and given some arbitrarily long spacelike geodesic, is it always ( or almost always, e.g. except for passing through a singularity) possible to find some open spacelike 3-region containing the geodesic (possibly an exceedingly skinny region around the geodesic, as long as it is an open region) that is quasi-euclidean in geometry?

There may be a nicer defintion, but what I mean by a quasi-euclidean spacelike 3-region is that one can find some coordinate system on a spacetime (4) region containing the 3-region where:

1) one coordinate is zero throughout the 3-region

2) the 3x3 submetric corresponding to the other coordinates is positive definite throughout the 3-region.

Though phrased in coordinates, I believe this definition is coordinate independent; a given spacelike 3-region either has or does not have this property. Note, also, that even in Minkowski spacetime, it is trivial to construct spacelike 3-surfaces that are not quasi-euclidean (start with one that is and add sufficient bumps, but still keep it spacelike; you then can't achieve positive definite metric throughout - some areas will have negative metric components, brought in from the overall psuedo-riemmanion metric of the spacetime; if you squeeze them out in one area, you'll pick them up in another, no matter how you pick coordinates - as long as they are legitimate coordinates).