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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: physics.stackexchange.com/q/83693 –  ramanujan_dirac May 25 at 13:53
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You may be interested in globally hyperbolic manifolds and initial value problem in general relativity en.wikipedia.org/wiki/Globally_hyperbolic_manifold –  cesaruliana May 26 at 0:38

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