Generalized Reeb Component
المؤلف:
Eliashberg, Y. M. and Thurston, W. P
المصدر:
Confoliations. Providence, RI: Amer. Math. Soc., 1998.
الجزء والصفحة:
...
7-7-2021
1826
Generalized Reeb Component
Given a compact manifold 
and a transversely orientable codimension-one foliation 
on 
which is tangent to 
, the pair 
is called a generalized Reeb component if the holonomy groups of all leaves in the interior 
are trivial and if all leaves of 
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 
admitting a generalized Reeb component also admits a nice vector field 
(Imanishi and Yagi 1976). Moreover, given a generalized Reeb component 
, 
is a fibration over 
.
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 
with foliation 
to be a submanifold 
of maximal dimension which is bounded by tori ![<span style=]()
{T_alpha}" src="https://mathworld.wolfram.com/images/equations/GeneralizedReebComponent/Inline16.gif" style="height:15px; width:25px" /> such that the orientation of these tori as leaves of 
is the same as (or simultaneously opposite to) their orientation as the boundary components of 
(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 
to admit a taut foliation.
REFERENCES:
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.
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