# Deductions about extending a pseudo-riemannian manifold

1. Jan 14, 2014

### PAllen

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.

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:

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

2. Jan 15, 2014

### PAllen

The answer is yes, the hoped for property is true.

The method is simply that for an argument that would use balls in Riemannian geometry, for pseudo-Rieamannian geometry you need an argument in terms of causal diamonds: region bounded by a past light cone from P1 with a future light cone from P2 (P1 in causal future of P2).

3. Jan 15, 2014

### Ben Niehoff

Analytic extensions are not unique. For example, consider the metric on some open region of a round sphere:

$$ds^2 = d\theta^2 + \sin \theta \; d\phi^2$$
We can consider $\theta \in (0,\pi/4)$, $\phi \in [0,2\pi)$ so that we're talking about a spherical cap. There are at least two distinct maximal extensions of this. We could extend to the usual 2-sphere, or we could extend to $RP^2$. The metric tensor is the same.

However, in $S^2$, geodesics through (0,0) intersect in two points, whereas in $RP^2$, geodesics through (0,0) intersect in one point.

I'm not sure how this relates to your question, because it's unclear to me what you're trying to achieve.

4. Jan 15, 2014

### PAllen

Unless I misunderstand you, this does not contradict what I'm proposing (and now think is true except for possible technical wording details) in the first two paragraphs of post #1 of this thread. Focus just on those (the rest is background and related questions).