Generalized Reeb Component

Given a compact manifold M and a transversely orientable codimension-one foliation F on M which is tangent to partialM, the pair (M,F) is called a generalized Reeb component if the holonomy groups of all leaves in the interior M^◦ are trivial and if all leaves of F are proper. Generalized Reeb components are obvious generalizations of Reeb components.

The introduction of the generalized version of the Reeb component facilitates the proof of many significant results in the theory of 3-manifolds and of foliations. It is well-known that generalized Reeb components are transversely orientable and that a manifold M admitting a generalized Reeb component also admits a nice vector field X (Imanishi and Yagi 1976). Moreover, given a generalized Reeb component (M,F), M^◦ is a fibration over S^1.

Like many notions in geometric topology, the generalized Reeb component can be presented in various contexts. One source describes a generalized Reeb component on a closed 3-manifold M with foliation F to be a submanifold N subset M of maximal dimension which is bounded by tori {T_alpha} such that the orientation of these tori as leaves of F is the same as (or simultaneously opposite to) their orientation as the boundary components of N (Eliashberg and Thurston 1998). Framed in this way, generalized Reeb components are shown to have deep connections to various notions in foliation theory, e.g., in presenting an existence criterion for a closed 3-manifold M to admit a taut foliation.

See also

Foliation, Foliation Leaf, Manifold, Nice Vector Field, Reeb Component, Reeb Foliation, Taut Foliation

This entry contributed by Christopher Stover

Explore with Wolfram|Alpha


Eliashberg, Y. M. and Thurston, W. P. Confoliations. Providence, RI: Amer. Math. Soc., 1998.Goodman, S. "Closed Leaves in Foliations of Codimension One." Comm. Math. Helv. 50, 383-388, 1975.Imanishi, H. and Yagi, K. "On Reeb Components." J. Math. Kyoto Univ. 16, 313-324, 1976.

Cite this as:

Stover, Christopher. "Generalized Reeb Component." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

Subject classifications