Question on submanifolds of a semi-riemannian manifold

[PAllen]

Arguably, this is pure mathematical question, but most discussions of semi-riemannian manifolds are in the context of physics, so I post here.

Can anyone state or point me to references discussing best known answers to the following:

Given an arbitrary Semi-Riemannian 4-manifold, and an arbitrary open 4-d subset of it, under what conditions on the subset (e.g. orientable, torsion free, not closed, metric condition ...?) is it possible to achieve the following (on the subset):

1) The subset can be covered with some family of non-intersecting 3-surfaces on which the induced metric is Euclidean flat.

2) The subset can be covered with some family of non-intersecting 2x1 Minkowski flat 3-manifolds.

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This is sort of a converse question the following example that the geometry of embedded surfaces can be very different from the geometry of the space they are embedded in: Flat Euclidean 3-space can be covered, except for one point, with family of non-intersecting 2-spheres.

 

[PAllen]

A follow up is, if one needs fairly strong metric conditions on the subset to achieve what I posed above, would the conditions be much weaker if loosened the requirements as follows:

1) The ability to find one embedded 3-surface meeting one of the conditions, rather than a covering of them.

2) The ability to find a 2-surface meeting the relevant condition containing 3 arbitrary (chosen) points have (a) spacelike relationship or (b) two with spacelike relationship, one with timelike to one of the others.

 

[PAllen]

Ok, I think I can answer these questions roughly, at least, by implicit function arguments.

The surfaces I seek above can be thought of as parts of coordinate systems, and the flatness conditions as coordinate conditions. One may impose 4 coordinate conditions in GR without loss of generality. The curvature tensor for a 2-surface has only one functionally independent component; for a 3-surface, 6 independent components.

Thus, it seems, one should generally (blah, blah degenerate cases) be able to find Euclidean flat or Minkowski flat (as appropriate) 2 surfaces containing any 3 chosen points. But you cannot generally find any flat 3-surfaces (you would need to impose 6 conditions, which is too many).

Thoughts?

 

[TrickyDicky]

Thus, it seems, one should generally (blah, blah degenerate cases) be able to find Euclidean flat or Minkowski flat (as appropriate) 2 surfaces containing any 3 chosen points. But you cannot generally find any flat 3-surfaces (you would need to impose 6 conditions, which is too many).

Thoughts?

I fail to see the motivation of this exercise. How does it relate to GR?

 

[PAllen]

I fail to see the motivation of this exercise. How does it relate to GR?

In the OP I noted it was really a mathematical question, but figured I would get faster answers here.

Its limited relevance to GR is roughly: we know you can't flatten curved space time by coordinate transform. To what degree can you partly flatten it? To what degree can you make particular coordinate surfaces flat? I think I have demonstrated that you can normally make 2-surfaces flat, but not 3-surfaces.