Foliage

The scrolling ( French feuilletage , eng. Foliation ) of a manifold is a term from the mathematical sub-area of differential topology . The topological theory of foliage was essentially founded by Georges Reeb .

One -dimensional foliation a -manifold you can of how the layers slate imagine ${\displaystyle 2}$ ${\displaystyle 3}$

A -dimensional foliation of a manifold is a decomposition of into disjoint, path-connected sets that look locally around every point like a layering of parallel -dimensional submanifolds. The elements are called the leaves of ; the leaves are not necessarily closed or even compact . ${\displaystyle k}$ ${\displaystyle {\mathcal {K}}=\{K_{\theta }\}}$ ${\displaystyle M}$ ${\displaystyle k}$ ${\displaystyle K_{\theta }}$ ${\displaystyle {\mathcal {K}}}$

definition

Map transition of a scrolled manifold

Be a smooth manifold . A partition of in disjoint path-connected sets is called foliation of when an Atlas exists (i. E. Is an open overlap and are Diffeomorphisms ), so that the image of each non-empty connection component from below in a plane is mapped. The elements are called the leaves of ${\displaystyle M}$ ${\displaystyle {\mathcal {K}}=\{K_{\theta }\}}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle {\mathcal {A}}=\{(U_{\eta },h_{\eta })\}}$ ${\displaystyle \{U_{\eta }\}}$ ${\displaystyle h_{\eta }:\,U_{\eta }\to \mathbb {R} ^{n}}$ ${\displaystyle K_{\theta }\cap U_{\eta }}$ ${\displaystyle h_{\eta }}$ ${\displaystyle k}$ ${\displaystyle \mathbb {R} ^{k}\times \{(x_{k+1},\dots ,x_{n})\}\subset \mathbb {R} ^{n}}$ ${\displaystyle K_{\theta }}$ ${\displaystyle {\mathcal {K}}}$

Examples

Puff pastry also has a similar structure to it.

• Be a non-vanishing vector field on , the form flow lines of a one-dimensional foliation. ${\displaystyle X}$ ${\displaystyle M}$ ${\displaystyle X}$
• In general, leaves do not form a submanifold globally . On the - Torus , consider the constant vector field . Each line of flux winds tightly around the torus. Thus, the topology of such a leaf does not match the topology of (This is also an example that not every subgroup of a Lie group is a Lies subgroup). ${\displaystyle 2}$ ${\displaystyle T^{2}=\mathbb {R} ^{2}/\mathbb {Z} ^{2}}$ ${\displaystyle (1,{\sqrt {2}})}$ ${\displaystyle \mathbb {R} ^{1}}$
• Be a bundle of fibers , then there is a foliage. ${\displaystyle p\colon M\to B}$ ${\displaystyle \{K_{b}\}:=\{p^{-1}(b):b\in B\}}$
• More generally, be a submersion , then there is a flip. An example of submersion that is not a bundle of fibers is . This provides foliation from , invariant under translation , the induced foliation on is the 2-dimensional orientable Reeb foliation . Furthermore, the foliation is also invariant under , in this case the induced foliation on the Möbius strip is the 2-dimensional non-orientable Reeb foliation. ${\displaystyle p\colon M\to B}$ ${\displaystyle \{K_{b}\}:=\{p^{-1}(b):b\in B\}}$ ${\displaystyle f(x,y)=(x^{2}-1)e^{y}}$ ${\displaystyle [-1,1]\times \mathbb {R} }$ ${\displaystyle (x,y)\to (x,y+1)}$ ${\displaystyle [-1,1]\times S^{1}}$ ${\displaystyle (x,y)\to (-x,y+1)}$
• Let F be a homeomorphism of a manifold, then the mapping torus of f has a foliation transverse to the fibers, the so-called suspension foliation. ${\displaystyle f\colon F\to F}$
• The Hopf fiber is a foliation in circles. From Vogt's theorem it follows that it also has a scroll in circles. ${\displaystyle S^{3}}$ ${\displaystyle \mathbb {R} ^{3}}$

Integrability

→ Main article : Frobenius theorem (differential topology)

In the examples above, a partition was not specified directly , but instead only one direction was specified at each point, and the question arose whether there is a foliation so that each leaf is tangent to the specified direction at each point. Often one finds similar situations in practice: A -dimensional distribution is given on a manifold . This is a -dimensional sub- bundle of tangent space . Whether there is a foliation for this distribution that is tangential to it can often be answered by Frobenius' theorem. ${\displaystyle {\mathcal {K}}}$ ${\displaystyle M}$ ${\displaystyle k}$ ${\displaystyle k}$

The Lie bracket of two vector fields, which are defined on a manifold, again results in a vector field on the manifold. Since each leaf of a foliation locally has the shape of a submanifold, it then follows that for any two vector fields that are tangential to (and which only have to be defined on this leaf) is also tangential again . The Frobenius theorem , however, also implies the reverse direction. ${\displaystyle K_{\theta }}$ ${\displaystyle {\mathcal {K}}=\{K_{\theta }\}}$ ${\displaystyle X,Y}$ ${\displaystyle K_{\theta }}$ ${\displaystyle [X,Y]}$ ${\displaystyle K_{\theta }}$

Theorem of Frobenius (after Ferdinand Georg Frobenius ): For a -dimensional distribution there is a tangential -dimensional scrolling if and only if for any vector fields that lie in, whose Lie bracket also forms a section in again . ${\displaystyle k}$ ${\displaystyle D}$ ${\displaystyle k}$ ${\displaystyle X,Y}$ ${\displaystyle D}$ ${\displaystyle [X,Y]}$ ${\displaystyle D}$

Bott's theorem provides a topological obstruction to the integrability of distributions .

Bott's theorem (after Raoul Bott ): If a -dimensional distribution has a tangential -dimensional foliation, then the ring produced by the Pontryagin classes of vanishes in dimensions . ${\displaystyle k}$ ${\displaystyle D}$ ${\displaystyle k}$ ${\displaystyle TM/D}$ ${\displaystyle Pont(TM/D)}$ ${\displaystyle d>2(n-k)}$

Existence proposition

Thurston's theorem (after William Thurston ): A closed smooth n-dimensional manifold has a smooth (n-1) -dimensional foliation if and only if its Euler characteristic is zero. If the Euler characteristic is zero, then every (n-1) -dimensional hyperplane field is homotopic to the tangent plane field of a smooth foliage.

Tight foliage

→ Main article : Tight foliage

There is an elaborated structure theory in codimension 1, especially for tight scrolling. These do not contain Reeb leaves and there is a Riemannian metric so that all leaves are minimal areas.

Leaves of surfaces

If F is a leafed closed surface , then F is either a torus or a Klein bottle and the foliage is either the suspension foliage of a homeomorphism or it consists of several (orientable or non-orientable) Reeb foliage. ${\displaystyle f\colon S^{1}\to S^{1}}$

Leaves of 3-manifolds

Codimension 1

Set of Novikov-Zieschang (after Sergei Novikov and Heiner Zieschang ): If there on a closed, orientable 3-manifold is a 2-dimensional foliation without Reeb components, then and all leaves are incompressible. ${\displaystyle M\not =S^{2}\times S^{1}}$ ${\displaystyle \pi _{2}(M)=0}$

Palmeira's theorem: If there is a 2-dimensional foliage without Reeb components on a closed, orientable 3-manifold , then the universal superimposition is diffeomorphic to and the raised foliage is a foliation of the foliage diffeomorphic to . ${\displaystyle M\not =S^{2}\times S^{1}}$ ${\displaystyle R^{3}}$ ${\displaystyle R^{3}}$ ${\displaystyle R^{2}}$

Gabai's theorem (after David Gabai ): Let M be a closed, irreducible 3-manifold with , then there is a 2-dimensional foliation on M without Reeb components. ${\displaystyle H_{2}(M)\not =0}$

Codimension 2

Epstein's theorem (after David Epstein ): Every foliation of a compact 3-manifold by circles is a Seifert fiber .

Theorem of Vogt (after Elmar Vogt ): If a 3-manifold has a foliation through circles, then every manifold created by removing finitely many points also has a (not necessarily differentiable) foliation through circles. ${\displaystyle M}$ ${\displaystyle M}$

Invariants of foliations

• Godbillon-Vey invariant
• Basic cohomology

See also

• Lamination (math)

literature

• R. Abraham, Jerrold E. Marsden , T. Ratiu: Manifolds, Tensor Analysis and Applications. , Springer-Verlag, Berlin 2003, ISBN 0-201-10168-8 , chap. 4.4
• A. Candel, L. Conlon, Foliations. I, II, American Mathematical Society, Providence, RI, 2000
• G. Hector, U. Hirsch, Introduction to the geometry of foliations. Part A, B, Friedr. Vieweg & Son, Braunschweig, 1981

Web links

• Foliations (Manifold Atlas)
• Thomas Vogel: Deformations of foliage (Yearbook of the Max Planck Society 2014)

Commons : Foliations  - collection of images, videos and audio files

swell

1. ^ Bott, Raoul: On a topological obstruction to integrability. 1970 Global Analysis (Proc. Sympos. Pure Math., Vol. XVI, Berkeley, Calif., 1968) pp. 127-131 Amer. Math. Soc., Providence, RI
2. ^ Thurston, WP: Existence of codimension-one foliations. Ann. of Math. (2) 104 (1976) no. 2, 249-268.
3. ↑ Hector, Gilbert; Hirsch, Ulrich: Introduction to the geometry of foliations. Part A. Foliations on compact surfaces, fundamentals for arbitrary codimension, and holonomy. Second edition. Aspects of Mathematics, 1st Friedr. Vieweg & Sohn, Braunschweig, 1986. xii + 234 pp. ISBN 3-528-18501-5
4. ↑ Novikov, SP: Топология слоений. Тр. Моск. мат. о-ва. 14 1965. 248-278.
5. ^ Palmeira, Carlos Frederico Borges: Open manifolds foliated by planes. Ann. Math. (2) 107 (1978) no. 1, 109-131.
6. ↑ Gabai, David: Foliations and the topology of 3-manifolds. J. Differential Geom. 18 (1983) no. 3, 445-503.
7. ^ David Epstein: Periodic flows on three manifolds. Ann. of Math. (2) 95 1972 66-82.
8. ↑ Elmar Vogt: A foliation of and other punctured 3-manifolds by circles. ${\displaystyle \mathbb {R} ^{3}}$Inst. Hautes Études Sci. Publ. Math. No. 1989, 69: 215-232.