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1.
Nicolas Ginoux Georges Habib 《Abhandlungen aus dem Mathematischen Seminar der Universit?t Hamburg》2008,78(1):69-90
We study a Killing spinor type equation on spin Riemannian flows. We prove integrability conditions and partially classify
those flows carrying non-trivial solutions.
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2.
We refine the cyclic cohomological apparatus for computing the Hopf cyclic cohomology of the Hopf algebras associated to infinite primitive Cartan–Lie pseudogroups, and for the transfer of their characteristic classes to foliations. The main novel feature is the precise identification as a Hopf cyclic complex of the image of the canonical homomorphism from the Gelfand–Fuks complex to the Bott complex for equivariant cohomology. This provides a convenient new model for the Hopf cyclic cohomology of the geometric Hopf algebras, which allows for an efficient transport of the Hopf cyclic classes via characteristic homomorphisms. We illustrate the latter aspect by indicating how to realize the universal Hopf cyclic Chern classes in terms of explicit cocycles in the cyclic cohomology of étale foliation groupoids. 相似文献
3.
Yuri A. Kordyukov 《Mathematische Nachrichten》2002,245(1):104-128
We consider a (hypo)elliptic pseudodifferential operator Ah on a closed foliated manifold (M,ℱ), depending on a parameterh > 0, of the form Ah = A+hmB, where A is a formally self–adjoint tangentially elliptic operator of orderμ > 0 with the nonnegative principal symbol and B is a formally self–adjoint classical pseudodi.erential operator of orderm > 0 on M with the holonomy invariant transversal principal symbol such that its principal symbol is positive, if μ < m, and its transversal principal symbol is positive, if μ ≥ m. We prove an asymptotic formula for the eigenvalue distribution function Nh(λ) of the operator Ah when h tends to 0 and λ is constant. 相似文献
4.
《Expositiones Mathematicae》2021,39(3):454-479
In a recent paper, the authors proved that no spin foliation on a compact enlargeable manifold with Hausdorff homotopy graph admits a metric of positive scalar curvature on its leaves. This result extends groundbreaking results of Lichnerowicz, Gromov and Lawson, and Connes on the non-existence of metrics of positive scalar curvature. In this paper we review in more detail the material needed for the proof of our theorem and we extend our non-existence results to non-compact manifolds of bounded geometry. We also give a first obstruction result for the existence of metrics with (not necessarily uniform) leafwise PSC in terms of the A-hat class in Haefliger cohomology. 相似文献
5.
Smectic liquid crystals are materials formed by stacking deformable, fluid layers. Although smectics prefer to have flat, uniformly-spaced layers, boundary conditions can impose curvature on the layers. Since the layer spacing and curvature are intertwined, the problem of finding minimal configurations for the layers becomes nontrivial. We discuss various topological and geometrical aspects of these materials and present recent progress on finding some exact layer configurations. We also exhibit connections to the study of certain embedded minimal surfaces and briefly summarize some important open problems. 相似文献
6.
For a Lie groupoid G with a twisting σ (a PU(H)-principal bundle over G), we use the (geometric) deformation quantization techniques supplied by Connes tangent groupoids to define an analytic index morphism in twisted K-theory. In the case the twisting is trivial we recover the analytic index morphism of the groupoid.For a smooth foliated manifold with twistings on the holonomy groupoid we prove the twisted analog of the Connes–Skandalis longitudinal index theorem. When the foliation is given by fibers of a fibration, our index coincides with the one recently introduced by Mathai, Melrose, and Singer.We construct the pushforward map in twisted K-theory associated to any smooth (generalized) map f:W→M/F and a twisting σ on the holonomy groupoid M/F, next we use the longitudinal index theorem to prove the functoriality of this construction. We generalize in this way the wrong way functoriality results of Connes and Skandalis when the twisting is trivial and of Carey and Wang for manifolds. 相似文献
7.
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
8.
A transversely homogeneous foliation is a foliation whose transverse model is a homogeneous space G/H. In this paper we consider the class of transversely homogeneous foliations
on a manifold M which can be defined by a family of 1-forms on M fulfilling the Maurer–Cartan equation of the Lie group G. This class includes as particular cases Lie foliations and certain homogeneous spaces foliated by points. We develop, for the foliations belonging to this class, a deformation theory for which both the foliation
and the model homogeneous space G/H are allowed to change. As the main result we show that, under some cohomological assumptions, there exist a versal space of deformations of finite dimension for the foliations of the class and when the manifold M is compact. Some concrete examples are discussed. 相似文献
9.
In Johnson and Smith (Indiana Univ Math J 44:45–85, 1995; Ann Global Anal Geometry 30:239–287, 2006; Proceedings of the VII
International Colloquium on Differential Geometry, 1994, World Scientific, pp. 81–98), the authors characterized the singular
set (discontinuities of the graph) of a volume-minimizing rectifiable section of a fiber bundle, showing that, except under
certain circumstances, there exists a volume-minimizing rectifiable section with the singular set lying over a codimension-3
set in the base space. In particular, it was shown that for 2-sphere bundles over 3-manifolds, a minimizer exists with a discrete
set of singular points. In this article, we show that for a 2-sphere bundle over a compact 3-manifold, such a singular point
cannot exist. As a corollary, for any compact 3-manifold, there is a C
1 volume-minimizing one-dimensional foliation. In addition, this same analysis is used to show that the examples, due to Pedersen
(Trans Am Math Soc 336:69–78, 1993), of potentially volume-minimizing rectifiable sections (rectifiable foliations) of the
unit tangent bundle to S
2n+1 are not, in fact, volume minimizing.
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10.
We prove generic semipositivity of the tangent bundle of a non-uniruled Calabi–Yau variety in positive characteristic. We also construct an example of a nef line bundle in characteristic zero, whose each reduction to positive characteristic is not nef. 相似文献