共查询到20条相似文献,搜索用时 296 毫秒
1.
In this paper we find smooth embeddings of solenoids in smooth foliations. We show that if a smooth foliation F of a manifold M contains a compact leaf L with H1(L;R) not equal to 0 and if the foliation is a product foliation in some saturated open neighborhood U of L, then there exists a foliation F′ on M which is C1-close to F, and F′ has an uncountable set of solenoidal minimal sets contained in U that are pairwise non-homeomorphic. If H1(L;R) is 0, then it is known that any sufficiently small perturbation of F contains a saturated product neighborhood. Thus, our result can be thought of as an instability result complementing the stability results of Reeb, Thurston and Langevin and Rosenberg. 相似文献
2.
César Camacho 《Topology and its Applications》2007,154(6):1032-1040
A closed, connected oriented three-manifold supporting a codimension one oriented smooth foliation with Morse singularities having more centers than saddles and without saddle connections is diffeomorphic to the three-sphere. The use of the Reeb Stability theorem in place of the Poincaré-Bendixson theorem paves the way to a three-dimensional version, for foliations with singularities of Morse type, of a classical result of Haefliger. Finally, we give an example of a codimension one C∞ foliation in the closed ball , with only one singularity which is of saddle type 2-2 and transverse to the boundary S3=∂B4. 相似文献
3.
A. Arbieto 《Topology and its Applications》2009,156(8):1491-1495
We show that a C0 codimension one foliation with C1 leaves F of a closed manifold is minimal if there are a foliation G transverse to F, and a diffeomorphism f preserving both foliations, such that every leaf of F intersects every leaf of G and f expands G. We use this result to study of Anosov actions on closed manifolds. 相似文献
4.
Anne Marie Torpe 《Journal of Functional Analysis》1985,61(1):15-71
The K-theory of the C1-algebra associated to C∞-foliations (V, F) of a manifold V in the simplest non-trivial case, i.e., dim V = 2, is studied. Since the case of the Kronecker foliation was settled by Pimsner and Voiculescu (J. Operator Theory4 (1980), 93–118), the remaining problem deals with foliations by Reeb components. The K-theory of for the Reeb foliation of S3 is also computed. In these cases the C1-algebra is obtained from simpler C1-algebras by means of pullback diagrams and short exact sequences. The K-groups are computed using the associated Mayer-Vietoris and six-term exact sequences. The results characterize the C1-algebra of the Reeb foliation of 2 uniquely as an extension of C(S1) by C(S1). For the foliations of 2 it is found that the K-groups count the number of Reeb components separated by stable compact leaves. A C∞-foliation of 2 such that K1(C1(2, F)) has infinite rank is also constructed. Finally it is proved, by explicit calculation using (M. Penington, “K-Theory and C1-Algebras of Lie Groups and Foliations,” D. Phil. thesis, Oxford, 1983), that the natural map is an isomorphism for foliations by Reeb components of 2 and S3. In particular this proves the Baum-Connes conjecture (P. Baum and A. Connes, Geometric K-theory for Lie groups, preprint, 1982; A. Connes, Proc. Symp. Pure Math.38 (1982), 521–628) when V = 2. 相似文献
5.
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
6.
We study smooth foliations on the solid torus S1×D2 having S1×{0} and S1×∂D2 as the only compact leaves and S1×{0} as singular set. We show that all other leaves can only be cylinders or planes, and give necessary conditions for the foliation to be a suspension of a diffeomorphism of the disc. 相似文献
7.
Yaning Wang 《Czechoslovak Mathematical Journal》2017,67(1):73-86
A necessary and sufficient condition for the Reeb vector field of a three dimensional non-Kenmotsu almost Kenmotsu manifold to be minimal is obtained. Using this result, we obtain some classifications of some types of (k, μ, v)-almost Kenmotsu manifolds. Also, we give some characterizations of the minimality of the Reeb vector fields of (k, μ, v)-almost Kenmotsu manifolds. In addition, we prove that the Reeb vector field of an almost Kenmotsu manifold with conformal Reeb foliation is minimal. 相似文献
8.
Alexander Vasil'ev 《Bulletin des Sciences Mathématiques》2005,129(10):831
We study one-parameter curves on the universal Teichmüller space T and on the homogeneous space M=DiffS1/RotS1 embedded into T. As a result, we deduce evolution equations for conformal maps that admit quasiconformal extensions and, in particular, such that the associated quasidisks are bounded by smooth Jordan curves. This approach allows us to understand the Laplacian growth (Hele-Shaw problem) as a flow in the Teichmüller space. 相似文献
9.
We investigate the equivariant cohomology of the natural torus action on a K-contact manifold and its relation to the topology of the Reeb flow. Using the contact moment map, we show that the equivariant cohomology of this action is Cohen–Macaulay, the natural substitute of equivariant formality for torus actions without fixed points. As a consequence, generic components of the contact moment map are perfect Morse-Bott functions for the basic cohomology of the orbit foliation ${{\mathcal F}}$ of the Reeb flow. Assuming that the closed Reeb orbits are isolated, we show that the basic cohomology of ${{\mathcal F}}$ vanishes in odd degrees, and that its dimension equals the number of closed Reeb orbits. We characterize K-contact manifolds with minimal number of closed Reeb orbits as real cohomology spheres. We also prove a GKM-type theorem for K-contact manifolds which allows to calculate the equivariant cohomology algebra under the nonisolated GKM condition. 相似文献
10.
In this paper we give a complete classification of simply connected homogeneous almost α-Kenmotsu three-manifolds M whose Ricci operator is invariant along the Reeb flow. We get this classification by using the Gaussian and the extrinsic curvature associated with the canonical foliation of M. 相似文献
11.
Let X be a C1 vectorfield on S2 = {(x, y, z)?R: x2 + y2 + z2 = 1} such that no open subset of S2 is the union of closed orbits of X. If X has only a finite number of singular orbits and satisfies one additional condition, then it is shown that X is topologically equivalent to a polynomial vectorfield. A corollary is that a foliation of the plane is topologically equivalent to a foliation by orbits of a polynomial vectorfield if and only if has only a finite number of inseparable leaves. 相似文献
12.
Amalendu Ghosh 《Results in Mathematics》2014,65(1-2):81-94
We show that if a compact K-contact metric is a gradient Ricci almost soliton, then it is isometric to a unit sphere S 2n+1. Next, we prove that if the metric of a non-Sasakian (κ, μ)-contact metric is a gradient Ricci almost soliton, then in dimension 3 it is flat and in higher dimensions it is locally isometric to E n+1 × S n (4). Finally, a couple of results on contact metric manifolds whose metric is a Ricci almost soliton and the potential vector field is point wise collinear with the Reeb vector field of the contact metric structure were obtained. 相似文献
13.
Domenico Perrone 《Mathematische Zeitschrift》2009,263(1):125-147
It is well known that a Hopf vector field on the unit sphere S
2n+1 is the Reeb vector field of a natural Sasakian structure on S
2n+1. A contact metric manifold whose Reeb vector field ξ is a harmonic vector field is called an H-contact manifold. Sasakian and K-contact manifolds, generalized (k, μ)-spaces and contact metric three-manifolds with ξ strongly normal, are H-contact manifolds. In this paper we study, in dimension three, the stability with respect to the energy of the Reeb vector
field ξ for such special classes of H-contact manifolds (and with respect to the volume when ξ is also minimal) in terms of Webster scalar curvature. Finally, we extend for the Reeb vector field of a compact K-contact (2n+1)-manifold the obtained results for the Hopf vector fields to minimize the energy functional with mean curvature correction.
Supported by funds of the University of Lecce and M.I.U.R.(PRIN). 相似文献
14.
Martin Ma?aj 《Indagationes Mathematicae》2005,16(2):237-250
Let G be an abelian group, S ⊆ G be a finite set, and T denote the multiplicative group of complex unitswith the invariant arc metric | arg(a/b)|. We will show that for a mapping ? : S → T to be ε-close on S to a character φ : G → T it is enough that ? be extendable to a mapping ¯f : (S U {1} US−1)n → T, where n is big enough and ¯f violates the homomorphy condition at most up to an arbitrary σ < min(ε, π/2). Moreover, n can be chosen uniformly, independently of G and both ? and ¯f, depending just on σ, ε and the number of elements of S. The proof is non-constructive, using the ultraproduct construction and Pontryagin duality, hence yielding no estimate on the actual size of n. As one of the applications we show that, for a vector u ∈ R q to be ε-close to some vector from the dual lattice H★ of a full rank integral point lattice H ⊆ ?q, it is enough for the scalar product ux to be δ-close (with δ < 1/3) to an integer for all vectors xH satisfying Σi|xi | < n, where n depends on δ, ε and q only. 相似文献
15.
The paper is concerned with the question of smoothness of the carrying simplex S for a discrete-time dissipative competitive dynamical system. We give a necessary and sufficient criterion for S being a C1 submanifold-with-corners neatly embedded in the nonnegative orthant, formulated in terms of inequalities between Lyapunov exponents for ergodic measures supported on the boundary of the orthant. This completes one thread of investigation occasioned by a question posed by M.W. Hirsch in 1988. Besides, amenable conditions are presented to guarantee the Cr (r?1) smoothness of S in the time-periodic competitive Kolmogorov systems of ODEs. Examples are also presented, one in which S is of class C1 but not neatly embedded, the other in which S is not of class C1. 相似文献
16.
Norimichi Hirano 《Journal of Differential Equations》2006,229(1):121-137
In this paper we prove the existence of nonstationary periodic solutions of delay Lotka-Volterra equations. In the proofs we use the S1-degree due to Dylawerski et al. [G. Dylawerski, K. Geba, J. Jodel, W. Marzantowicz, An S1-equivariant degree and the Fuller index, Ann. Polon. Math. 63 (1991) 243-280]. 相似文献
17.
18.
D. V. Bolotov 《Mathematical Notes》1998,63(5):575-581
C∞-foliations of codimension 1 on compact Riemannian 3-manifolds are studied. New classes of foliations, namely hyperbolic,
elliptic, and parabolic foliations, are considered. Examples of such foliations are presented. In particular, aC∞-metric of nonnegative sectional curvature onS
3 such that the Reeb foliation is parabolic with respect to this metric is constructed. Analytic 3-manifolds with sectional
curvature of constant sign admitting parabolic foliations are classified.
Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 651–659, May, 1998.
The author wishes to express his thanks to Professor A. A. Borisenko for his supervision, and to Yu. A. Nikolaevskii for useful
advice in the process of preparing the present paper. 相似文献
19.
Given a continuous function f:X→? on a topological space X, its level set f ?1(a) changes continuously as the real value a changes. Consequently, the connected components in the level sets appear, disappear, split and merge. The Reeb graph of f summarizes this information into a graph structure. Previous work on Reeb graph mainly focused on its efficient computation. In this paper, we initiate the study of two important aspects of the Reeb graph, which can facilitate its broader applications in shape and data analysis. The first one is the approximation of the Reeb graph of a function on a smooth compact manifold M without boundary. The approximation is computed from a set of points P sampled from M. By leveraging a relation between the Reeb graph and the so-called vertical homology group, as well as between cycles in M and in a Rips complex constructed from P, we compute the H 1-homology of the Reeb graph from P. It takes O(nlogn) expected time, where n is the size of the 2-skeleton of the Rips complex. As a by-product, when M is an orientable 2-manifold, we also obtain an efficient near-linear time (expected) algorithm for computing the rank of H 1(M) from point data. The best-known previous algorithm for this problem takes O(n 3) time for point data. The second aspect concerns the definition and computation of the persistent Reeb graph homology for a sequence of Reeb graphs defined on a filtered space. For a piecewise-linear function defined on a filtration of a simplicial complex K, our algorithm computes all persistent H 1-homology for the Reeb graphs in $O(n n_{e}^{3})$ time, where n is the size of the 2-skeleton and n e is the number of edges in K. 相似文献
20.
For a Lagrangian submanifold M of S 6 with nearly Kaehler structure, we provide conditions for a canonically induced almost contact metric structure on M by a unit vector field, to be Sasakian. Assuming M contact metric, we show that it is Sasakian if and only if the second fundamental form annihilates the Reeb vector field ξ, furthermore, if the Sasakian submanifold M is parallel along ξ, then it is the totally geodesic 3-sphere. We conclude with a condition that reduces the normal canonical almost contact metric structure on M to Sasakian or cosymplectic structure. 相似文献