# Foliation with isometric leaves.

1. May 25, 2014

### center o bass

Suppose one has a foliation of a manifold $M$ with codimension one leaves that are all isometric. What is such a foliation called?

All I have been able to find online is something called quasi-isometric foliations, which does not seem to coincide with my definition above.

2. Jun 26, 2014

### Greg Bernhardt

I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?

3. Jun 26, 2014

### Matterwave

I only know this from the context of General Relativity. If the hypersurfaces are isometric to each other, and are space like, this means one can define a global time coordinate on such hypersurfaces. The integral curves of the velocity vector of this global time coordinate can be used to define the isometry (maybe diffeomorphism in GR) between different hypersurfaces. Such surfaces are then called Cauchy surfaces, and the space-time is said to be globally hyperbolic.

Probably this is not what you wanted...but it's all I can think of.

4. Jun 27, 2014

### Geometry_dude

I don't know whether there already is a word, but if you were to approach me on the street and start talking to me about isometric foliations, there's two things I would have in mind. The first one is what you described, that is a foliation of a manifold into mutually isometric leaves. The next one would be two (pseudo-)Riemannian manifolds $(M,g),(M',g')$ that are foliated and there exists an isometry between the leaves of the two with respect to the pullback metric. With respect to this metric, the leaves are assumed to be pseudo-Riemannian as well. Note that this neither implies that $(M,g)$ and $(M',g')$ are isometric, nor that the leaves are mutually isometric. Actually, the more I think about this, the more it appears that the latter is best described by the word "isometric foliation".
It follows that this would be a bad choice of terminology. However, both foliations and metrics are omnipresent in differential geometry, so there's probably a word... I spent some time looking but I really couldn't find anything.
To be safe I would simply stick to "a foliation with isometric leaves" or define your own terminology.