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1.
Riemannian foliations are characterized as those foliations whose holonomy pseudogroup consists of local isometries of a Riemannian
manifold. Their main structural features are well understood since the work of Molina. In this paper we analyze the more general
concept of equicontinuous pseudogroup of homeomorphisms, which gives rise to the notion of equicontinuous foliated space.
We show that equicontinuous foliated spaces have structural properties similar to those known for Riemannian foliations: the
universal covers of their leaves are in the same quasi-isometry class, leaf closures are homogeneous spaces, and the holonomy
pseudogroup is indeed given by local isometries. 相似文献
2.
M. Zhitomirskii 《Proceedings of the Steklov Institute of Mathematics》2007,259(2):281-293
The paper is devoted to the classification of nonsingular and singular plane curve germs with respect to the group of local
diffeomorphisms preserving the foliation of the plane by the phase curves of a fixed vector field, either nonsingular or singular.
We define the multiplicity of a pair consisting of a plane curve and a vector field and prove an analog of the Tougeron theorem
on finite determinacy. It leads, almost immediately, to a number of classification results; a part of them is contained in
the work.
Dedicated to Vladimir Igorevich Arnold on the occasion of his 70th birthday 相似文献
3.
For fibred boundary and fibred cusp metrics, Hausel, Hunsicker, and Mazzeo identified the space of L2 harmonic forms of fixed degree with the images of maps between intersection cohomology groups of an associated stratified space obtained by collapsing the fibres of the fibration at infinity onto its base. In the present paper, we obtain a generalization of this result to situations where, rather than a fibration at infinity, there is a Riemannian foliation with compact leaves admitting a resolution by a fibration. If the associated stratified space (obtained now by collapsing the leaves of the foliation) is a Witt space and if the metric considered is a foliated cusp metric, then no such resolution is required. 相似文献
4.
M. Zhitomirskii 《Proceedings of the Steklov Institute of Mathematics》2007,259(1):281-293
The paper is devoted to the classification of nonsingular and singular plane curve germs with respect to the group of local
diffeomorphisms preserving the foliation of the plane by the phase curves of a fixed vector field, either nonsingular or singular.
We define the multiplicity of a pair consisting of a plane curve and a vector field and prove an analog of the Tougeron theorem
on finite determinacy. It leads, almost immediately, to a number of classification results; a part of them is contained in
the work. 相似文献
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6.
Björn Gustafsson Jacqueline Mossino Colette Picard 《Annali di Matematica Pura ed Applicata》1992,162(1):87-104
Let \(\Omega = \Omega _0 \backslash \bar \Omega _1\) be a regular annulus inR N and \(\phi :\bar \Omega \to R\) be a regular function such that φ=0 on ?Ω0, φ=1 on ?Ω1 and ▽φ ≠ 0. Let Kn be the subset of functions v ε W1,p (Ω) such that v=0 on ?Ω0, v=1 on ?Ω1, v=(unprescribed) constant on n given level surfaces of φ. We study the convergence of sequences of minimization problems of the type $$Inf\left\{ {\int\limits_\Omega {\frac{1}{{a_n \circ \phi }}G(x,(a_n \circ \phi )\nabla v)dx;v \in K_n } } \right\},$$ where an ε L∞ (0,1) and G: (x, ζ) ε Ω × RN → G(x, ζ εR is convex with respect to ξ and verifies some standard growth conditions. 相似文献
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Given a generalized Robertson-Walker spacetime whose warping function verifies a certain convexity condition, we classify strongly stable spacelike hypersurfaces with constant mean curvature. More precisely, we will show that given a closed, strongly stable spacelike hypersurface of with constant mean curvature H, if the warping function ? satisfying ?″?max{H?′,0} along M, then Mn is either maximal or a spacelike slice Mt0={t0}×F, for some t0∈I. 相似文献
13.
《Indagationes Mathematicae》2021,32(1):121-150
We prove a Thom isomorphism theorem for differential forms in the setting of transverse Lie algebra actions on foliated manifolds and foliated vector bundles. 相似文献
14.
In this article, we generalize known integral formulae (due to Brito–Langevin–Rosenberg, Ranjan and the second author) for
foliations of codimension 1 or unit vector fields and obtain an infinite series of such formulae involving invariants of the
Weingarten operator of a unit vector field, of the Jacobi operator in its direction, and their products. We write several
such formulae explicitly, on locally symmetric spaces as well as on arbitrary Riemannian manifolds where they involve also
covariant derivatives of the Jacobi operator. We work also with foliations of codimension 1 (or vector fields) which admit
“good” (in a sense) singularities. 相似文献
15.
Rafael López 《Monatshefte für Mathematik》2008,154(4):289-302
In this paper we study surfaces in Euclidean 3-space foliated by pieces of circles that satisfy a Weingarten condition of
type aH + bK = c, where a,b and c are constant, and H and K denote the mean curvature and the Gauss curvature respectively. We prove that such a surface must be a surface of revolution,
one of the Riemann minimal examples, or a generalized cone.
Authors’ address: Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain 相似文献
16.
Shinsuke Yorozu 《Israel Journal of Mathematics》1986,56(3):349-354
We give a generalization of the result obtained by C. Currás-Bosch. We consider the Av-operator associated to a transverse Killing fieldν on a complete foliated Riemannian manifold (M, ℱ, g). Under a certain assumption, we prove that, for eachx ∈M, (Av)
x
belongs to the Lie algebra of the linear holonomy group ψv(x). A special case of our result, the version of the foliation by points, implies the results given by B. Kostant (compact
case) and C. Currás-Bosch (non-compact case). 相似文献
17.
Yuri A. Kordyukov 《Journal of Fixed Point Theory and Applications》2010,7(2):385-399
First, we review the notion of a Poisson structure on a noncommutative algebra due to Block, Getzler and Xu, and we introduce
a notion of a Hamiltonian vector field on a noncommutative Poisson algebra. Then we describe a Poisson structure on a noncommutative
algebra associated with a transversely symplectic foliation and construct a class of Hamiltonian vector fields associated
with this Poisson structure. 相似文献
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19.
We study ${({\mathcal{F}}, {\mathcal{G}})}$ -harmonic maps between foliated Riemannian manifolds ${(M, {\mathcal{F}}, g)}$ and ${(N, {\mathcal{G}}, h)}$ i.e. smooth critical points ? : M → N of the functional ${E_T (\phi ) = \frac{1}{2} \int_M \| d_T \phi \|^2 \,d \, v_g}$ with respect to variations through foliated maps. In particular we study ${({\mathcal{F}}, {\mathcal{G}})}$ -harmonic morphisms i.e. smooth foliated maps preserving the basic Laplace equation Δ B u = 0. We show that CR maps of compact Sasakian manifolds preserving the Reeb flows are weakly stable ${({\mathcal{F}}, {\mathcal{G}})}$ -harmonic maps. We study ${({\mathcal{F}}, {\mathcal{G}}_0 )}$ -harmonic maps into spheres and give foliated analogs to Solomon’s (cf., J Differ Geom 21:151–162, 1985) results. 相似文献
20.
Sébastien Alvarez 《Comptes Rendus Mathematique》2012,350(11-12):621-626
We prove in this Note that there is, for some foliated bundles, a bijective correspondence between Garnett?s harmonic measures and measures on the fiber that are stationary for some probability measure on the holonomy group. As a consequence, we show the uniqueness of the harmonic measure in the case of some foliations transverse to projective fiber bundles. 相似文献