Hypersurfaces with vanishing extrinsic curvature

• arild
In summary, the extrinsic curvature of hypersurfaces can be a complex constraint to deal with in GR. Attempting to find references relevant to this specific constraint in the physics literature has been unsuccessful. Totally geodesic hypersurfaces are a specific type of hypersurface that are relevant to this constraint.
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.

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.

coalquay404 said:
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).
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.

Totally geodesic hyperslices

arild said:
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.

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

Chris Hillman

1. What are hypersurfaces with vanishing extrinsic curvature?

Hypersurfaces with vanishing extrinsic curvature are mathematical surfaces that have a curvature of zero when viewed from the outside. This means that the surface does not bend or curve in any direction when observed from the outside.

2. What is the significance of hypersurfaces with vanishing extrinsic curvature?

These hypersurfaces have important implications in the study of geometry and physics. They are often used in the field of general relativity, where they represent the boundaries of spacetime regions with no gravitational forces.

3. How are hypersurfaces with vanishing extrinsic curvature different from other surfaces?

Hypersurfaces with vanishing extrinsic curvature are different from other surfaces in that they have a constant curvature of zero, while other surfaces may have varying degrees of curvature.

4. Can hypersurfaces with vanishing extrinsic curvature exist in the real world?

Yes, hypersurfaces with vanishing extrinsic curvature can exist in the real world. In fact, they are commonly observed in nature, such as in the shape of soap bubbles or the surface tension of liquids.

5. How are hypersurfaces with vanishing extrinsic curvature used in practical applications?

Hypersurfaces with vanishing extrinsic curvature have practical applications in various fields, such as computer graphics and medical imaging. They are also used in the study of minimal surfaces, which have important applications in material science and engineering.

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