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I am interested on getting some hints on how Foliations Theory of Manifolds can be used fruitfully on General Relativity. I just started my Ph.D on Mathematics this semester focusing on studying Holomorphic foliations on projective manifolds. I just came across some texts which related the more general Foliation theory to Gen. Rel.

Is there research going on that mixes both areas? Can someone direct me to introductory papers and textbooks?

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Look at Sean Carol's note's linked in this question. This is one example when foliation of manifolds can be used to guess the form of a metric using the Frobenius theorem: – user7757 May 25 '14 at 13:53
You may be interested in globally hyperbolic manifolds and initial value problem in general relativity – cesaruliana May 26 '14 at 0:38

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up vote 4 down vote accepted

I'll elaborate on the things mentioned in the comments. That being said, this will refer mostly to very specific foliations and not much of a general theory.

First, since manifolds in GR have a timelike and a spacelike component, it's always worth keeping track of those. Then foliations arise very naturally trying to see whether or not there is a reasonable initial value problem formulation for the Einstein field equations. To see this, consider that they usually come in 4-dimensional notation $$ G_{\mu\nu} + \Lambda g_{\mu\nu} = \kappa T_{\mu\nu},$$ which can be derived from varying the corresponding Einstein-Hilbert action w.r.t. $g_{\mu\nu}$ $$S= \int_V \mathrm{d}^4x \sqrt{-g} \frac{\kappa}{2} \{ R +\mathcal{L} \} + \int_{\partial V} \text{stuff}, $$ $R$ being the Ricci scalar, $\mathcal{L}$ the Lagrangian density for matter (and cosmological constant $\Lambda$), $g$ the determinant of the metric - the boundary terms I alluded to, but the details are slightly messy, don't help much, but nevertheless I think that as a mathematician you like seeing that they still play a role in principle.

The term initial value formulation already tells the source of possible problems: "Initial" hints at time and the usual 4D version mixes those up heavily. Now the foliation comes in: We can find an initial value formulation (and a Hamiltonian density formulation) as long as the manifold proves to be globally hyperbolic. The technical definition (as found on Wikipedia in the link) boils down to the ability to find, in a sense, that the manifold can be decomposed as $$ \mathbb{R}\times M_3, $$ for some three-manifolds $M_3$ and the real line basically giving the time direction. The leaves of the foliation are more or less and very roughly equal-time hypersurfaces.

Second, you can consider different kinds of foliations, as well. One example would be to separate a certain notion of time of the space part (say, $S_1\times M_3$). The other certain geometric parts within the 3-manifold - like separating out the spherically symmetric part as two-sphere leaves, giving rise to the quite ubiquitous $\mathrm{d} \Omega$ in line elements.

Skimming through the tables of contents and indices of the books by Choquet-Bruhat, Hawking & Ellis, Beem, Ehrlich & Easley and O'Neill, however, I couldn't find hints at a very general theory of this (fibre, leaf, foliation are not in the indices). The best I could come up with is one very specific type of foliation that can usually be found in many textbooks (e.g. Gourgoulhon (linked the version on the arxiv)) called the 3+1-decomposition needed for initial value formulations and therefore most of the work on numerical relativity. Of course, that still leaves a bit of a choice how to do this particular foliation, but it's already very focussed on a certain application and maybe not as general as you might be interested in. Still, Gourgoulhon's book/lecture notes has a lot to offer from an applied point of view.

This book by Reinhart offers some references to different applications in technical papers. Still, as far as I could tell, specific foliations for certain metrics, and a lot to the above mentioned problems.

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