Foliations of Manifolds  

In General
* Idea: A p-dimensional foliation of an n-dimensional manifold M is a decomposition of M as a union of parallel submanifolds (leaves) of dimension p.
* Leaves: Each leaf in a foliation is an embedded submanifold of dimension p.
* 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!
> Related topics: see embeddings and immersions; extrinsic curvature [and extremal surfaces]; gauge transformations; Hypersurface; metric decomposition.
> Special cases: see 3D manifolds; FLRW spacetime.
> Online resources: see Wikipedia page.

Foliations of Riemannian Manifolds
* Singular Riemannian foliation: A singular foliation is called a singular Riemannian foliation if every geodesic that is perpendicular to one leaf is perpendicular to every leaf it meets; A typical example is the partition of a complete Riemannian manifold into orbits of an isometric action.
@Singular Riemannian foliation: Alexandrino et al DG&A(13) [intro].

Foliations of Lorentzian Manifolds > s.a. 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 CQG(07)-a0709 [with respect to a static Killing vector, uniqueness].
@ Lightlike foliations: Bolós MPCPS(05)gq.
@ Constant mean curvature: Christodoulou & Francaviglia GRG(79) [Einstein spacetimes admitting a Gaussian foliation]; Andersson et al JGP(97)gq/96 [2+1 globally hyperbolic]; Rendall HPA(96)gq [conjectures], CMP(97)gq/96 [with 2 Killing vector fields], gq/96-proc [results]; Andersson & Iriondo gq/97; Isenberg & Rendall CQG(98)gq/97 [counterexamples]; Gowdy gq/01 [Minkowski and Schwarzschild spacetimes]; Malec & Ó Murchadha PRD(03)gq, gq/03, PRD(09)-a0903 [Schwarzschild spacetime]; Klainerman & Rodnianski a0801 [breakdown]; Martín-Moruno AIP(12)-a1201 [in our region of spacetime].
@ Other types: Berger et al AP(97)gq [T2 isometry]; Henkel AIP(02)gq/01, JMP(02)gq/01, JMP(02)gq/01 [with prescribed mean curvature]; Müller a0904/TAMS [globally hyperbolic manifolds, nice foliations]; García JMP(13)-a1212 [conformally flat leaves].
@ And singularities: Alcubierre CQG(03) [focusing singularities and gauge shocks].

Related Topic: Webs
* Web: A family of foliations of a manifold in general positions.
* Orthogonal web: In an n-dimensional manifold with metric, a set of n paiwise transversal and orthogonal foliations of connected submanifolds of codimension 1.
@ Web: Chern BAMS(82) [geometry]; Zakharevich mp/00 [Veronese webs]; Grifone & Salem ed-01; Fleischhack MN(04)mp/03 [parallel transport], CMP(04) [degenerate webs].
> Online resources: see Wikipedia page.

References
@ General: Reinhart 83; Bejancu & Farran 06 [and geometric structures].
@ 3D manifolds: Shields T&A(06) [equivalence classes of foliations]; Calegari 07.
@ Manifolds that can / cannot be leaves of foliations: Cantwell & Conlon Top(87) [every surface is a leaf]; Attie & Hurder Top(96) [manifolds that cannot].


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