In mathematics, Reeb sphere theorem, named after Georges Reeb, states that A singularity of a foliation F is of Morse type if in its small neighborhood all leaves of the foliation are level sets of a Morse function, being the singularity a critical point of the function.
The singularity is a center if it is a local extremum of the function; otherwise, the singularity is a saddle.
, is tightly connected with the manifold topology.
ind p = min ( k , n − k )
{\displaystyle \operatorname {ind} p=\min(k,n-k)}
, where k is the index of the corresponding critical point of a Morse function.
A Morse foliation F on a manifold M is a singular transversely oriented codimension one foliation of class
with isolated singularities such that: This is the case
be a closed oriented connected manifold of dimension
-transversely oriented codimension one foliation
with a non empty set of singularities all of them centers.
It is a consequence of the Reeb stability theorem.
In 1978, Edward Wagneur generalized the Reeb sphere theorem to Morse foliations with saddles.
He showed that the number of centers cannot be too much as compared with the number of saddles, notably,
: He obtained a description of the manifold admitting a foliation with singularities that satisfy (1).
be a compact connected manifold admitting a Morse foliation
, Finally, in 2008, César Camacho and Bruno Scardua considered the case (2),
This is possible in a small number of low dimensions.
be a compact connected manifold and