Foliations of Manifolds

In General
* Result: Every complete 2-manifold can occur as the leaf of a foliation of a compact 3-manifold, but not every complete 3-manifold can occur as the leaf of a foliation of a 4D one!
* Webs: Families of foliations of a manifold in general positions.
> Related topics: see embeddings; extrinsic curvature [including extremal surfaces]; gauge; Hypersurface; metric decomposition.
> Special cases: see 3D manifolds; FRW spacetime.

Foliations of Lorentzian Manifolds > initial value formulation of general relativity.
* Spacelike case: A spacetime foliation follows from (but is weaker than) a choice of time or gauge in an initial value formulation; Every foliation gives a curve in the space of Riemannian geometries (+ a lapse function + a shift vector); The choice of an appropriate foliation is important in numerical relativity.
@ Spacelike and timelike foliations of spacetime: in Lau CQG(96)gq/95; Walschap JGP(99) [spacelike, properties]; Harris & Low CQG(01)gq [shape of space]; Husain et al PRD(02)gq/01 [spherical, spatially flat]; Bolós GRG(07)gq/05 [stability between foliations].
@ Special spacelike foliations: Delphenich gq/02 [proper time foliations]; Sánchez & Senovilla a0709 [wrt a static Killing vector, juniqueness].
@ Lightlike foliations: Bolós MPCPS(05)gq.
@ Constant mean curvature: Christodoulou & Francaviglia GRG(79) [Einstein spacetimes admitting a Gaussian foliation]; Andersson et al gq/96 [2+1 globally hyperbolic]; Rendall HPA(96)gq [conjectures], CMP(97)gq/96 [with 2 Killing vector fields], gq/96-in [results]; Andersson & Iriondo gq/97; Isenberg & Rendall CQG(98)gq/97 [counterexamples]; Gowdy gq/01 [Minkowski and Schwarzschild]; Malec & Ó Murchadha PRD(03)gq, gq/03 [Schwarzschild]; Klainerman & Rodnianski a0801 [breakdown].
@ Other types: Berger et al AP(97)gq [T² isometry]; Henkel gq/01, JMP(02)gq/01, JMP(02)gq/01 [prescribed mean curvature].
@ And singularities: Alcubierre CQG(03) [focusing singularities and gauge shocks].

References
@ General: Reinhart 83; Bejancu & Farran 05 [and geometric structures].
@ 3D manifolds: Shields T&A(06) [equivalence classes of foliations].
@ Manifolds that cannot be leaves of foliations: Attie & Hurder Top(96).
@ Webs: Chern BAMS(82) [geometry]; Zakharevich mp/00 [Veronese webs]; Grifone & Salem ed-01; Fleischhack MN(04)mp/03 [parallel transport], CMP(04) [degenerate webs].

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