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Lilia Rosati 《Topology and its Applications》2012,159(5):1388-1403
Let M be a smooth manifold and let F be a codimension one, C∞ foliation on M, with isolated singularities of Morse type. The study and classification of pairs (M,F) is a challenging (and difficult) problem. In this setting, a classical result due to Reeb (1946) [11] states that a manifold admitting a foliation with exactly two center-type singularities is a sphere. In particular this is true if the foliation is given by a function. Along these lines a result due to Eells and Kuiper (1962) [4] classifies manifolds having a real-valued function admitting exactly three non-degenerate singular points. In the present paper, we prove a generalization of the above mentioned results. To do this, we first describe the possible arrangements of pairs of singularities and the corresponding codimension one invariant sets, and then we give an elimination procedure for suitable center-saddle and some saddle-saddle configurations (of consecutive indices).In the second part, we investigate if other classical results, such as Haefliger and Novikov (Compact Leaf) theorems, proved for regular foliations, still hold true in presence of singularities. At this purpose, in the singular set, Sing(F) of the foliation F, we consider weakly stable components, that we define as those components admitting a neighborhood where all leaves are compact. If Sing(F) admits only weakly stable components, given by smoothly embedded curves diffeomorphic to S1, we are able to extend Haefliger?s theorem. Finally, the existence of a closed curve, transverse to the foliation, leads us to state a Novikov-type result. 相似文献
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A. Mafra 《Topology and its Applications》2010,157(13):2141-2144
We study codimension one smooth foliations with singularities on closed manifolds. We assume that the singularities are nondegenerate (of Bott-Morse type) in the sense of Scárdua and Seade (2009) [9] and prove a version of Thurston-Reeb stability theorem in terms of a component of the singular set: If all singularities are of center type and the foliation exhibits a compact leaf with trivial Cohomology group of degree one or a codimension ?3 component of the singular set with trivial Cohomology group of degree one then the foliation is compact and stable. 相似文献
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Percy Fernández-Sánchez Jorge Mozo-Fernández Hernán Neciosup 《Journal of Differential Equations》2014
In this paper, we will construct a pre-normal form for germs of codimension one holomorphic foliation having a particular type of separatrix, of cuspidal type. As an application, we will explain how this form could be used in order to study the analytic classification of the singularities via the projective holonomy, in the generalized surface case. We will also give an application to the analytic classification of singularities, and a sufficient condition, in the quasi-homogeneous, three-dimensional case, for these foliations to be of generalized surface type. 相似文献
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I. A. Mel'nikova 《Mathematical Notes》1998,63(6):760-763
In this paper foliations determined by Morse forms on compact manifolds are considered. An inequality involving the number of connected components of the set formed by noncompact leaves, the number of homologically independent compact leaves, and the number of singular points of the corresponding Morse form is obtained.Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 862–865, June, 1998.The author wishes to thank Professor A. S. Mishchenko for his interest in this work and stimulating discussions.This research was partially supported by the Russian Foundation for Basic Research under grant No. 96-01-00276. 相似文献
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Paul A. Schweitzer S. J. 《Annals of Global Analysis and Geometry》1991,9(1):77-81
Every smooth closed manifold of dimension 4 or greater that has a smooth codimension one foliation, has such aC
1 foliation whose leaves are minimal hypersurfaces for someC
1 Riemannian metric. 相似文献
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Janez Mrcun 《Proceedings of the American Mathematical Society》1997,125(5):1561-1570
We prove an equivariant version of the Reeb-Thurston stability theorem for foliations invariant under an action of a discrete group.
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The existence problem for attractors of foliations with transverse linear connection is investigated. In general foliations with transverse linear connection do not admit attractors. Sufficient conditions are found for the existence of a global attractor that is a minimal set. An application to transversely similar pseudo-Riemannian foliations is obtained. The global structure of transversely similar Riemannian foliations is described. Different examples are constructed. 相似文献
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Christian Pappas 《Transactions of the American Mathematical Society》1996,348(8):3065-3083
We give a method for constructing all of the extensions of an immersion, and determine the CW structure and diffeomorphism type of each.
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Taro Asuke 《Transactions of the American Mathematical Society》1996,348(5):1939-1958
Transversely flat conformal foliations with good transverse invariant measures are Riemannian in the sense. In particular, transversely similar foliations with good measures are transversely Riemannian as transversely -foliations.
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The question of the existence of a proper closed densely defined suboperator whose domain contains a given linear space is answered. Such suboperators with codimension one domains are described. The motivation for studying this topic comes from the theory of extremal extensions of (non-densely defined) positive operators. 相似文献