Codimension one foliations with Bott-Morse singularities I

Details Codimension one foliations with Bott-Morse singularities I Codimension one foliations with Bott-Morse singularities I

2006-08-23

arxiv.org/abs/math/0608585v1

Mathematics

Differential Geometry

3 figures

Scientific paper

We study codimension one (transversally oriented) foliations $\fa$ on oriented closed manifolds $M$ having non-empty compact singular set $\sing(\fa)$ which is locally defined by Bott-Morse functions. We prove that if the transverse type of $\fa$ at each singular point is a center and $\fa$ has a compact leaf with finite fundamental group or a component of $\sing(\fa)$ has codimension $\ge 3$ and finite fundamental group, then all leaves of $\fa$ are compact and diffeomorphic, $\sing(\fa)$ consists of two connected components, and there is a Bott-Morse function $f:M \to [0,1]$ such that $f\colon M \setminus \sing(\fa) \to (0,1)$ is a fiber bundle defining $\fa$ and $\sing(\fa) = f^{-1}(\{0,1\})$. This yields to a topological description of the type of leaves that appear in these foliations, and also the type of manifolds admiting such foliations. These results unify, and generalize, well known results for cohomogeneity one isometric actions and a theorem of Reeb for foliations with Morse singularities of center type. In this case each leaf of $\fa$ is a sphere fiber bundle over each component of $\sing(\fa)$.

Affiliated with

Also associated with

No associations

Rating

Codimension one foliations with Bott-Morse singularities I does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Codimension one foliations with Bott-Morse singularities I, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Codimension one foliations with Bott-Morse singularities I will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-110253