Reeb sphere theorem

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

A closed oriented connected manifold Mⁿ 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 $L\setminus p$ 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 $n\ge 2$ . 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 $c>s\ge 0.$

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, $c\le s+2$ . So there are exactly two cases when $c>s$ :

(1)

c=s+2, \,

(2)

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 $c\le s+2$ . 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 $s = c + 1$ , then

$n=2,4,8$ or $16$ ,

$M^n$ is an Eells–Kuiper manifold.

References

↑ 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 .
↑ Wagneur, E. (1978), "Formes de Pfaff à singularités non dégénérées", Annales de l'Institut Fourier (in French) 28 (3): xi, 165–176, MR 511820 .
↑ 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 .