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.
foliation of a manifold
a compact leaf with finite holonomy group.
(also called invariant), in which all the leaves are compact with finite holonomy groups.
is a covering map with a finite number of sheets and, for each
is homeomorphic to a disk of dimension k and is transverse to
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.
[2] This is the case of codimension one, singular foliations
The Reeb local stability theorem also has a version for a noncompact codimension-1 leaf.
[3][4] An important problem in foliation theory is the study of the influence exerted by a compact leaf upon the global structure of a foliation.
For certain classes of foliations, this influence is considerable.
, codimension one foliation of a closed manifold
with finite fundamental group, then all the leaves of
are compact, with finite fundamental group.
is transversely orientable, then every leaf of
is the total space of a fibration
This theorem holds true even when
is a foliation of a manifold with boundary, which is, a priori, tangent on certain components of the boundary and transverse on other components.
[5] In this case it implies Reeb sphere theorem.
Reeb Global Stability Theorem is false for foliations of codimension greater than one.
[6] However, for some special kinds of foliations one has the following global stability results: Theorem:[7] Let
be a complete conformal foliation of codimension
has a compact leaf with finite holonomy group, then all the leaves of
are compact with finite holonomy group.
be a holomorphic foliation of codimension
in a compact complex Kähler manifold.
has a compact leaf with finite holonomy group then every leaf of
is compact with finite holonomy group.