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 [T^{2} 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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