Codimension one foliations with Bott-Morse singularities I

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We study codimension one (transversally oriented) foliations on oriented closed manifolds M having non-empty compact singular set () which is locally defined by Bott-Morse functions. We prove that if the transverse type of at each singular point is a center and has a compact leaf with finite fundamental group or a component of () has codimension ≥ 3 and finite fundamental group, then all leaves of are compact and diffeomorphic, () consists of two connected components, and there is a Bott-Morse function f:M → [0,1] such that f M () → (0,1) is a fiber bundle defining and () = 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 is a sphere fiber bundle over each component of ().