Reeb stability for noncompact leaves

Open manifolds which are nonrealizable as leaves

“…In these several years, many people have published their results on the qualitative theory of foliations, mainly of class C 2 and of codimension one. These results can be more or less interpreted as those on the realizability or non-realizability of manifolds as leaves having a certain qualitative property (e. g. Sondow [15], Nishimori [10,11,12], Cantwell-Conlon [1,2,3], Tsuchiya [17,18,19], Phillips-Sullivan [13], Inaba [7,8], Takamura [16], and so on). From this point of view, one may naturally ask the existence of open manifolds which can never be realized as leaves of foliations of any closed manifold.…”

Molino theory for matchbox manifolds

“…The distinction between the holonomy group Germ(Φ 0 , x) being trivial, and it being locally trivial, may seem technical, but this distinction is related to fundamental dynamical properties of the foliation F P of S P . For example, it is a key concept in the generalizations of the Reeb stability theorem from compact leaves to the non-compact case for codimension-one foliations, as discussed in the works of Sacksteder and Schwartz [58] and Inaba [36,37]. The nomenclature "locally trivial" was introduced by Inaba [36,37].…”

“…For example, it is a key concept in the generalizations of the Reeb stability theorem from compact leaves to the non-compact case for codimension-one foliations, as discussed in the works of Sacksteder and Schwartz [58] and Inaba [36,37]. The nomenclature "locally trivial" was introduced by Inaba [36,37]. As we see below, this distinction is also important for the study of the dynamics of weak solenoids.…”

Reeb-Thurston stability for symplectic foliations

“…Generalizations to noncompact leaves of the Reeb stability theorem were studied before (see [1,6] and the references therein). The goal in these works is to produce saturated neighborhoods of embedded leaves (also called proper leaves in foliation literature) which are isomorphic to the flat bundle (7).…”

“…The goal in these works is to produce saturated neighborhoods of embedded leaves (also called proper leaves in foliation literature) which are isomorphic to the flat bundle (7). For this, finiteness of the holonomy group is not sufficient [6]. Results are known to hold only in low dimensions [1], for compact ambient manifolds, and with extra restrictions on the topology of the leaf.…”