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];
Gerhardt a1902
[spacetimes asymptotic to open FLRW spacetimes].

@ __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* pairwise
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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