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A counterexample to smooth leafwise Hodge decomposition for general foliations and to a type of dynamical trace formulas

Christopher Deninger, Wilhelm Singhof (2001)

Annales de l’institut Fourier

We construct a two dimensional foliation with dense leaves on the Heisenberg nilmanifold for which smooth leafwise Hodge decomposition does not hold. It is also shown that a certain type of dynamical trace formulas relating periodic orbits with traces on leafwise cohomologies does not hold for arbitrary flows.

A few remarks on the geometry of the space of leaf closures of a Riemannian foliation

Małgorzata Józefowicz, R. Wolak (2007)

Banach Center Publications

The space of the closures of leaves of a Riemannian foliation is a nice topological space, a stratified singular space which can be topologically embedded in ℝ k for k sufficiently large. In the case of Orbit Like Foliations (OLF) the smooth structure induced by the embedding and the smooth structure defined by basic functions is the same. We study geometric structures adapted to the foliation and present conditions which assure that the given structure descends to the leaf closure space. In Section...

A finiteness theorem for Riemannian submersions

Paweł G. Walczak (1992)

Annales Polonici Mathematici

Given some geometric bounds for the base space and the fibres, there is a finite number of conjugacy classes of Riemannian submersions between compact Riemannian manifolds.

A note on generalized flag structures

Tomasz Rybicki (1998)

Annales Polonici Mathematici

Generalized flag structures occur naturally in modern geometry. By extending Stefan's well-known statement on generalized foliations we show that such structures admit distinguished charts. Several examples are included.

A spectral estimate for the Dirac operator on Riemannian flows

Nicolas Ginoux, Georges Habib (2010)

Open Mathematics

We give a new upper bound for the smallest eigenvalues of the Dirac operator on a Riemannian flow carrying transversal Killing spinors. We derive an estimate on both Sasakian and 3-dimensional manifolds, and partially classify those satisfying the limiting case. Finally, we compare our estimate with a lower bound in terms of a natural tensor depending on the eigenspinor.

A Weitzenbôck formula for the second fundamental form of a Riemannian foliation

Paolo Piccinni (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Si considera la seconda forma fondamentale α di foliazioni su varietà riemanniane e si ottiene una formula per il laplaciano ∇ 2 ⁡ α - Se ne deducono alcune implicazioni per foliazioni su varietà a curvatura costante.

Algebroid nature of the characteristic classes of flat bundles

Jan Kubarski (1998)

Banach Center Publications

The following two homotopic notions are important in many domains of differential geometry: - homotopic homomorphisms between principal bundles (and between other objects), - homotopic subbundles. They play a role, for example, in many fundamental problems of characteristic classes. It turns out that both these notions can be - in a natural way - expressed in the language of Lie algebroids. Moreover, the characteristic homomorphisms of principal bundles (the Chern-Weil homomorphism [K4], or the...

An invariant of nonpositively curved contact manifolds

Thilo Kuessner (2011)

Open Mathematics

We define an invariant of contact structures and foliations (on Riemannian manifolds of nonpositive sectional curvature) which is upper semi-continuous with respect to deformations and thus gives an obstruction to the topology of foliations which can be approximated by isotopies of a given contact structure.

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