# Hypersurfaces with vanishing extrinsic curvature

1. Nov 10, 2006

### arild

Could anyone share insights/results/references on hypersurfaces with vanishing extrinsic curvature?
In particular, I would be interested in results related to existence (do they always exist, if not when do they exist?) and procedures for constructing them from the background geometry.

2. Nov 13, 2006

### coalquay404

It might help if you tell us a little bit about why you want to know. For example, are you interested in applying this to physics? The most important area in physics where the extrinsic curvature of hypersurfaces is important is general relativity (specifically, the 3+1 approach to GR).

There are literally hundreds of papers out there in the GR literature which are relevant to your question. Do a search at the ArXiv for them. In case you're interested, GR becomes extremely simple when you deal with such hypersurfaces (you'll hear them referred to as maximal or moment-of-symmetry hypersurfaces) since the constraint equations in GR decouple when you restrict your choices to maximal hypersurfaces.

3. Nov 13, 2006

### arild

Sorry if I was too brief in my original post. Yes, my problem is GR-related. As I understand it, maximal hypersurfaces refer to hypersurfaces where the trace of the extrinsic curvature vanishes. I see that my question was a little vague, but it relates to hypersurfaces where the extrinsic curvature tensor vanishes (K_{ab} = 0), not only its trace. Since this constraint is much harder than requiring the trace to vanish, my two primary questions are:
1) What are the conditions that a spacetime geometry must satisfy in order to admit such hypersurfaces?
2) If they exist, how can they be constructed?

I have so far been unable to find references in the physics literature that deal with this specific constraint. Any such reference would be very helpful.

Btw, since my original post, I found that hypersurfaces where the extrinsic curvature tensor vanishes are referred to as totally geodesic in the mathematics literature.

4. Dec 1, 2006

### Chris Hillman

Totally geodesic hyperslices

You will probably be interested by a nifty characterization of the EFE which uses this concept. See Frankel, Gravitational Curvature.

Chris Hillman