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 $s$, specifically c − s, is tightly connected with the manifold topology.

We denote ind p = min(k, n − k), the index of a singularity $p$, 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 $M^{n}$ be a closed oriented connected manifold of dimension $n\geq 2$. Assume that $M^{n}$ admits a $C^{1}$-transversely oriented codimension one foliation $F$ with a non empty set of singularities all of them centers. Then the singular set of $F$ consists of two points and $M^{n}$ is homeomorphic to the sphere $S^{n}$.

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

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

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

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

• $n=2,4,8$ or $16$,
• $M^{n}$ 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, arXiv:math/0611395, doi:10.1090/S0002-9939-08-09371-4, MR 2425748.