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Tao Li 《Proceedings of the American Mathematical Society》2002,130(8):2471-2477
Suppose a manifold has a codimension one, transversely orientable foliation without holonomy, and is a leaf. We give a simple, purely topological proof of the theorem that is a normal subgroup containing the commutator subgroup of .
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In order to understand the linearization problem around a leaf of a singular foliation, we extend the familiar holonomy map from the case of regular foliations to the case of singular foliations. To this aim we introduce the notion of holonomy transformation. Unlike the regular case, holonomy transformations cannot be attached to classes of paths in the foliation, but rather to elements of the holonomy groupoid of the singular foliation. 相似文献
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We generalize Frobenius singular theorem due to Malgrange, for a large class of codimension one holomorphic foliations on
singular analytic subsets of ℂ
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This research was partially supported by Pronex. 相似文献
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G. Katz 《Commentarii Mathematici Helvetici》2002,77(1):39-77
For a closed 1-form with Morse singularities, Calabi discovered a simple global criterion for the existence of a Riemannian metric in which is harmonic. For a codimension 1 foliation , Sullivan gave a condition for the existence of a Riemannian metric in which all the leaves of are minimal hypersurfaces. The conditions of Calabi and Sullivan are strikingly similar. If a closed form has no singularities, then both criteria are satisfied and, for an appropriate choice of metric, is harmonic and the associated foliation is comprised of minimal leaves. However, when has singularities, the foliation is not necessarily minimal.? We show that the Calabi condition enables one to find a metric in which is harmonic and the leaves of the foliation are minimal outside a neighborhood U of the -singular set. In fact, we prove the best possible result of this type: we construct families of metrics in which, as U shrinks to the singular set, the taut geometry of the foliation outside U remains stable. Furthermore, all compact leaves missing U are volume minimizing cycles in their homology classes. Their volumes are controlled explicitly.
Received: January 24, 2000 相似文献
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Luís Gustavo Mendes 《Bulletin of the Brazilian Mathematical Society》2000,31(2):127-143
We introduce numerical invariants of holomorphic singular foliations under bimeromorphic transformations of surfaces. The basic invariant is a foliated version of the Kodaira dimension of compact complex manifolds.The author was supported by CNPq-Brazil in 1998 and Conseil Régional de Bourgogne in 1999. 相似文献
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Science China Mathematics - In this paper, we investigate the smoothness of invariant manifolds and foliations for random dynamical systems with nonuniform pseudo-hyperbolicity in Hilbert spaces.... 相似文献
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Alexander Lytchak 《Geometriae Dedicata》2010,149(1):379-395
We prove that an isometric action of a Lie group on a Riemannian manifold admits a resolution preserving the transverse geometry
if and only if the action is infinitesimally polar. We provide applications concerning topological simplicity of several classes
of isometric actions, including polar and variationally complete ones. All results are proven in the more general case of
singular Riemannian foliations. 相似文献
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Robert A. Wolak 《Proceedings of the American Mathematical Society》2000,128(5):1543-1545
Basic forms for a transversely integrable singular Riemannian foliation with compact leaves are in one-to-one correspondence with ``Weyl"-invariant differential forms on a generalized section of the foliation.
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Patrick Gilmer 《Mathematische Annalen》1993,295(1):643-659
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We give a necessary and sufficient condition for a submanifold with parallel focal structure to give rise to a global foliation of the ambient space by parallel and focal manifolds. We show that this is a singular Riemannian foliation with complete orthogonal transversals. For this object we construct an action on the transversals that generalizes the Weyl group action for polar actions.
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Alexandrino Marcos M. Inagaki Marcelo K. de Melo Mateus Struchiner Ivan 《Annals of Global Analysis and Geometry》2022,61(3):593-619
Annals of Global Analysis and Geometry - We describe a local model for any singular Riemannian foliation in a neighborhood of a closed saturated submanifold of a singular stratum. Moreover, we... 相似文献
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