Suppose we have a pseudo-riemannian 4 manifold S (sometimes also called a Minkowskian manifold) that is without boundary, and not simply connected. Suppose there is at least one pseudo-riemannian 4 manifold M (also without boundary) that has S as a regular submanifold, preserving all geometry (pseudo-riemannian metric) and topology of S.(adsbygoogle = window.adsbygoogle || []).push({});

Suppose in S, there are a pair of spacelike geodesics that do not intersect in S, but their extension in M does intersect. Further, the intersection occurs at a boundary of S in M. I would like to conclude that if this is true for one M as described, then it is true for any M (any pseudo-riemannian 4-manifold, without boundary, containing S as a regular submanifold and preserving all of its geometry and topology).

A difficulty for pseudo-riemannian manifolds is that a geodesic of zero interval can still be topologically a line (a null geodesic). There can also be topological lines that are not geodesics that have invariant interval of zero (null paths). Thus, arguments about interval being zero or tending to zero don't tell you anything about intersection without additional reasoning.

Questions:

1) Is my hoped for conclusion true? If so, what is a strategy for establishing it?

The specific question I have concerns geometries of interest in General Relativity, where I posted the specific version of this question and got no response:

https://www.physicsforums.com/showthread.php?t=732569

And these wonderments of mine were triggered by the following paper which shows that my particular example of interest (SC exterior geometry) can be extended in some surprising ways, with very different topology than Kruskal (by which I mean the maximal analytic extension; the following paper uses it only as a style of coordinates):

http://arxiv.org/abs/0910.5194

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# Deductions about extending a pseudo-riemannian manifold

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