Reeb stability theorem

In mathematics, Reeb stability theorem, named after Georges Reeb, asserts that if one leaf of a codimension-one foliation is closed and has finite fundamental group, then all the leaves are closed and have finite fundamental group.

Reeb local stability theorem

Theorem: Let

F

{\displaystyle F}

be a

C

1

{\displaystyle C^{1}}

, codimension

k

{\displaystyle k}

foliation of a manifold

M

{\displaystyle M}

and

L

{\displaystyle L}

a compact leaf with finite holonomy group. There exists a neighborhood

U

{\displaystyle U}

of

L

{\displaystyle L}

, saturated in

F

{\displaystyle F}

(also called invariant), in which all the leaves are compact with finite holonomy groups. Further, we can define a retraction

π

:

U

→

L

{\displaystyle \pi :U\to L}

such that, for every leaf

L

′

⊂

U

{\displaystyle L'\subset U}

,

π

|

L

′

:

L

′

→

L

{\displaystyle \pi |_{L'}:L'\to L}

is a covering map with a finite number of sheets and, for each

y

∈

L

{\displaystyle y\in L}

,

π

−

1

(

y

)

{\displaystyle \pi ^{-1}(y)}

is homeomorphic to a disk of dimension k and is transverse to

F

{\displaystyle F}

. The neighborhood

U

{\displaystyle U}

can be taken to be arbitrarily small.

The last statement means in particular that, in a neighborhood of the point corresponding to a compact leaf

with finite holonomy, the space of leaves is Hausdorff.

Under certain conditions the Reeb local stability theorem may replace the Poincaré–Bendixson theorem in higher dimensions. This is the case of codimension one, singular foliations

(

M

n

,

F

)

{\displaystyle (M^{n},F)}

, with

n

≥

3

{\displaystyle n\geq 3}

, and some center-type singularity in

S

i

n

g

(

F

)

{\displaystyle Sing(F)}

.

The Reeb local stability theorem also has a version for a noncompact codimension-1 leaf.

Reeb global stability theorem

An important problem in foliation theory is the study of the influence exerted by a compact leaf upon the global structure of a foliation.