Isolated Ends of Open Leaves of Codimension-One Foliations

Nishimori, Toshiyuki

doi:10.1093/qmath/26.1.159

Cited by 20 publications

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“…Notable advances included Hector's work on classification and examples, starting with his Thesis and subsequent developments of its themes [132,133,135,138]; Lamoureux's work on holonomy and "captured leaves" [200,201,202,203]; Moussu's study of foliations almost without holonomy [239,240]; Nishimori's study of the asymptotic properties and growth of leaves in foliations [245,246,247,248]; and Plante's study of the relation between growth of leaves and the fundamental groups of the ambient manifolds.…”

Section: Foliation Dynamics In Codimension Onementioning

confidence: 99%

Classifying foliations

Hurder¹

2009

Foliations, Geometry, and Topology

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Abstract. We give a survey of the approaches to classifying foliations, starting with the Haefliger classifying spaces and the various results and examples about the secondary classes of foliations. Various dynamical properties of foliations are introduced and discussed, including expansion rate, local entropy, and orbit growth rates. This leads to a decomposition of the foliated space into Borel or measurable components with these various dynamical types. The dynamical structure is compared with the classification via secondary classes.1991 Mathematics Subject Classification. Primary 22F05, 37C85, 57R20, 57R32, 58H05, 58H10; Secondary 37A35, 37A55, 37C35 .

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Section: Foliation Dynamics In Codimension Onementioning

confidence: 99%

Classifying foliations

Hurder¹

2009

Foliations, Geometry, and Topology

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“…C'est un ensemble saturé en général non fermé de ^. On a E(F) = L(F) si, et seulement si, F est une feuille propre, au sens de [1, 8,9].…”

Section: Notationsunclassified

“…Un bout, au sens de [2,4,8], d'une feuille non fermée F est noté s ; il n'est pas nécessairement isolé dans l'espace des bouts de F [2,4,8].…”

Section: Notationsunclassified

“…L f ensemble-limite du bout s d'une feuille est noté L(e) ; par définition [8,10], on a L(e) = \ \ ^ où {^{qigE est …”

Section: Notationsunclassified

“…L'orientabilité transverse de ^ permet enfin de donner un sens [8] Démonstration. -Puisque l'ensemble L(£) saturé non vide du corollaire 1 est compact, il contient un ensemble minimal compact K; soit T un segment d'une feuille de ^ qui rencontre K. Si K ne se réduit pas à une feuille compacte, K est soit la variété X, soit un ensemble exceptionnel [1,10].…”

Section: L(s) = E(s)unclassified

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Quelques remarques sur les bouts de feuilles

Lamoureux¹

1977

Annales De L’institut Fourier

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Foliations

Fuks¹

1982

J Math Sci

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The survey is based on works on the theory of foliations reviewed in RZhMatematika during 1970-1979. The basic topics are the classification of foliations, characteristic classes, the qualitative theory of foliations (holonomy, growth of leaves, etc.), and special classes of foliations (compact foliations, Riemannian foliations, etc.). I. General TheoryThe theory of foliations has existed as an independent mathematical discipline for about 30 years, The first major work in this area is the dissertation of Reeb [302] which contains a number of deep theorems together with the basic definitions. During the fifties and sixties the number of works on the theory of foliations was relatively small, but some of these works were very important, for example, the works of Haefliger [137], Novikov [19], and Wood [388]. In the seventies interest in foliations rose considerably; the principal stimuli were the theory of characteristic classes of foliations which arose in 1971 and the works of Thurston of 1974-1976 on the existence and classification of foliations. Beginning in 1970 about 400 works on the theory of foliations were reviewed in RZhMatmatika; the present survey is written on the basis of these works~ Among these works specially devoted to foliations there are the book of Tamura[355] reeently published in Russian translation, the survey paper [211], more restricted survey papers [24,27, 47, 120,173,187, 311, 320, 3541, and the popular papers [69, 341, 385]. Among other works of general character we mention the story of the sources and the basic problems of foliations written by the founder [303] and the list of problems compiled by Schweitzer [329].i. Basic Concepts. Let X be a smooth n-dimensional manifold without boundary (smoothness everywhere means C =~ unless otherwise specified), and let 0 -< p -< n. We say that on X there is given a smooth foliation of dimension p if X is equipped with a partition into (linearly) connected subsets F~ possessing the following property: Each point x 6 X is covered by a chart ~: U ~ IR n from the structural atlas of the manifold X such that the components (relative to the linear connectedness) of the intersections UflF= are mapped by ~ onto pdimensional planes of the space R n parallel to the plane of the first p coordinate axes. The sets Fc~ are called leaves of the foliation, and X itself is called the total manifold. The number n-p is called the codimension of the foliation; the eodimension of the foliation ~ is denoted by codim ~-.These definitions (as the majority of definitions below) have obvious complex versions. They can also be extended to manifolds with boundary, while here there are two concurrent possibilities: it is possible to require that the leaves be transverse to the boundary, and it is possible to require that the components of the boundary be leaves. In the first case the foliation is called transversal to the boundary, while in the second case (in which the codimension of the foliation must be equal to i) the foliation is trivial on the boundary. Exampl...

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Isolated Ends of Open Leaves of Codimension-One Foliations

Cited by 20 publications

References 0 publications

Classifying foliations

Classifying foliations

Quelques remarques sur les bouts de feuilles

Foliations

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