# Reeb sphere theorem

In mathematics, Reeb sphere theorem, named after Georges Reeb, states that

A closed oriented connected manifold M n that admits a singular foliation having only centers is homeomorphic to the sphere Sn and the foliation has exactly two singularities.

## Morse foliation

A singularity of a foliation F is of Morse type if in its small neighborhood all leaves of the foliation are levels 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.

The number of centers c and the number of saddles ${\displaystyle s}$, specifically c − s, is tightly connected with the manifold topology.

We denote ind p = min(kn − k), the index of a singularity ${\displaystyle p}$, where k is the index of the corresponding critical point of a Morse function. In particular, a center has index 0, index of a saddle is at least 1.

A Morse foliation F on a manifold M is a singular transversely oriented codimension one foliation of class C2 with isolated singularities such that:

• each singularity of F is of Morse type,
• each singular leaf L contains a unique singularity p; in addition, if ind p = 1 then ${\displaystyle L\setminus p}$ is not connected.

## Reeb sphere theorem

This is the case c > s = 0, the case without saddles.

Theorem:[1] Let ${\displaystyle M^{n}}$ be a closed oriented connected manifold of dimension ${\displaystyle n\geq 2}$. Assume that ${\displaystyle M^{n}}$ admits a ${\displaystyle C^{1}}$-transversely oriented codimension one foliation ${\displaystyle F}$ with a non empty set of singularities all of them centers. Then the singular set of ${\displaystyle F}$ consists of two points and ${\displaystyle M^{n}}$ is homeomorphic to the sphere ${\displaystyle S^{n}}$.

It is a consequence of the Reeb stability theorem.

## Generalization

More general case is ${\displaystyle c>s\geq 0.}$

In 1978, E. 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, ${\displaystyle c\leq s+2}$. So there are exactly two cases when ${\displaystyle c>s}$:

(1) ${\displaystyle c=s+2,}$
(2) ${\displaystyle c=s+1.}$

He obtained a description of the manifold admitting a foliation with singularities that satisfy (1).

Theorem:[2] Let ${\displaystyle M^{n}}$ be a compact connected manifold admitting a Morse foliation ${\displaystyle F}$ with ${\displaystyle c}$ centers and ${\displaystyle s}$ saddles. Then ${\displaystyle c\leq s+2}$. In case ${\displaystyle c=s+2}$,

• ${\displaystyle M}$ is homeomorphic to ${\displaystyle S^{n}}$,
• all saddles have index 1,
• each regular leaf is diffeomorphic to ${\displaystyle S^{n-1}}$.

Finally, in 2008, C. Camacho and B. Scardua considered the case (2), ${\displaystyle c=s+1}$. Interestingly, this is possible in a small number of low dimensions.

Theorem:[3] Let ${\displaystyle M^{n}}$ be a compact connected manifold and ${\displaystyle F}$ a Morse foliation on ${\displaystyle M}$. If ${\displaystyle s=c+1}$, then

• ${\displaystyle n=2,4,8}$ or ${\displaystyle 16}$,
• ${\displaystyle M^{n}}$ is an Eells–Kuiper manifold.

## References

1. ^ Reeb, Georges (1946), "Sur les points singuliers d'une forme de Pfaff complètement intégrable ou d'une fonction numérique", C. R. Acad. Sci. Paris (in French), 222: 847–849, MR 0015613.
2. ^ Wagneur, E. (1978), "Formes de Pfaff à singularités non dégénérées", Annales de l'Institut Fourier (in French), 28 (3): xi, 165–176, MR 511820.
3. ^ Camacho, César; Scárdua, Bruno (2008), "On foliations with Morse singularities", Proceedings of the American Mathematical Society, 136 (11): 4065–4073, MR 2425748, arXiv:, doi:10.1090/S0002-9939-08-09371-4.