Leaf prescriptions for closed 3-manifolds

Cantwell, John; Conlon, Lawrence

doi:10.1090/s0002-9947-1978-0645738-9

Cited by 19 publications

(2 citation statements)

References 16 publications

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“…Each leaf of F is an orientable surface of infinite genus and its end set is the Cantor set consisting of non planar ends. By the classifying theorem ( [7], [9]) for non-compact surfaces (see also [1]), all leaves are diffeomorphic.…”

Section: Examplesmentioning

confidence: 99%

Codimension one minimal foliations and the fundamental groups of leaves

Yokoyama¹,

Tsuboi²

2008

Ann. inst. Fourier

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Section: Examplesmentioning

confidence: 99%

Codimension one minimal foliations and the fundamental groups of leaves

Yokoyama¹,

Tsuboi²

2008

Ann. inst. Fourier

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“…(Hector [150] also presented an example of such a foliation which moreover possesses certain other interesting properties.) In [61] it is proved that any two-dimensional surface of finite growth is realized as a leaf of a smooth foliation on a closed, three-dimensional manifold (moreover, it can be arranged that this foliation not have exceptional minimal sets). In [59] it is proved that a surface homeomorphic to I% 2 \ K, where K is the Cantor set, can also be a leaf of a foliation on a closed, three-dimensional manifold, but this leaf must have exponential growth.…”

Section: Foliations Of Codimensionmentioning

confidence: 99%

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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Analytic foliations and the theory of levels

Cantwell

Conlon

1983

Math. Ann.

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Leaf prescriptions for closed 3-manifolds

Cited by 19 publications

References 16 publications

Codimension one minimal foliations and the fundamental groups of leaves

Codimension one minimal foliations and the fundamental groups of leaves

Foliations

Analytic foliations and the theory of levels

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