Structure of a leaf of some codimension one riemannian foliation

Krystyna Bugajska

Displaying similar documents to “Structure of a leaf of some codimension one riemannian foliation”

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

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Si considera la seconda forma fondamentale $\alpha$ di foliazioni su varietà riemanniane e si ottiene una formula per il laplaciano $\nabla^{2}\alpha$ - Se ne deducono alcune implicazioni per foliazioni su varietà a curvatura costante.

De Rham decomposition theorems for foliated manifolds

Robert A. Blumenthal, James J. Hebda (1983)

Annales de l'institut Fourier

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We prove that if $M$ is a complete simply connected Riemannian manifold and $F$ is a totally geodesic foliation of $M$ with integrable normal bundle, then $M$ is topologically a product and the two foliations are the product foliations. We also prove a decomposition theorem for Riemannian foliations and a structure theorem for Riemannian foliations with recurrent curvature.

Tenseness of Riemannian flows

Hiraku Nozawa, José Ignacio Royo Prieto (2014)

Annales de l’institut Fourier

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We show that any transversally complete Riemannian foliation $ℱ$ of dimension one on any possibly non-compact manifold $M$ is tense; namely, $M$ admits a Riemannian metric such that the mean curvature form of $ℱ$ is basic. This is a partial generalization of a result of Domínguez, which says that any Riemannian foliation on any compact manifold is tense. Our proof is based on some results of Molino and Sergiescu, and it is simpler than the original proof by Domínguez. As an application, we generalize...

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

Małgorzata Józefowicz, R. Wolak (2007)

Banach Center Publications

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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....

Warped compact foliations

Szymon M. Walczak (2008)

Annales Polonici Mathematici

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The notion of the Hausdorffized leaf space $\tilde{}$ of a foliation is introduced. A sufficient condition for warped compact foliations to converge to $\tilde{}$ is given. Moreover, a necessary condition for warped compact Hausdorff foliations to converge to $\tilde{}$ is shown. Finally, some examples are examined.

Correspondence between diffeomorphism groups and singular foliations

Tomasz Rybicki (2012)

Annales Polonici Mathematici

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It is well-known that any isotopically connected diffeomorphism group G of a manifold determines a unique singular foliation $ℱ_{G}$ . A one-to-one correspondence between the class of singular foliations and a subclass of diffeomorphism groups is established. As an illustration of this correspondence it is shown that the commutator subgroup [G,G] of an isotopically connected, factorizable and non-fixing $C^{r}$ diffeomorphism group G is simple iff the foliation $ℱ_{[G, G]}$ defined by [G,G] admits no proper...

Extending regular foliations

J. W. Smith (1969)

Annales de l'institut Fourier

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A $p$ -dimensional foliation $F$ on a differentiable manifold $M$ is said to extend provided there exists a $(p + 1)$ -dimensional foliation $F^{'}$ on $M$ with $F \subset F^{'}$ . Our main result asserts that if $M$ and $F$ extends over relatively compact subsets of $M$ .

Natural liftings of foliations to the $r$ -tangent bunde

Mikulski, Włodzimierz M.

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Let $F$ be a $p$ -dimensional foliation on an $n$ -manifold $M$ , and $T^{r} M$ the $r$ -tangent bundle of $M$ . The purpose of this paper is to present some reltionship between the foliation $F$ and a natural lifting of $F$ to the bundle $T^{r} M$ . Let $L_{q}^{r} (F)$ $(q = 0, 1, \dots, r)$ be a foliation on $T^{r} M$ projectable onto $F$ and $L_{q}^{r} = {L_{q}^{r} (F)}$ a natural lifting of foliations to $T^{r} M$ . The author proves the following theorem: Any natural lifting of foliations to the $r$ -tangent bundle is equal to one of the liftings $L_{0}^{r}, L_{1}^{r}, \dots, L_{n}^{r}$ . The exposition is clear and well organized. ...

Foliations by complex manifolds involving the complex Hessian

Julian Ławrynowicz, Jerzy Kalina, Masami Okada

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SummaryIn 1979 the second named author proved, in a joint paper with J. Ławrynowicz, the existence of a foliation of a bounded domain in $ℂ^{n}$ by complex submanifolds of codimension k+p-1, connected in some sense with a real (1,1) C³-form of rank k and the pth power of the complex Hessian of a C³-function u with im u plurisubharmonic and the property that for every leaf of this foliation the restricted functions im u, re u and $(\partial / \partial z_{j}) i m u$ , $(\partial / \partial z_{j}) r e u$ are pluriharmonic and holomorphic, respectively.Now the...

Geometry of manifolds which admit conservation laws

David E. Blair, Alexander P. Stone (1971)

Annales de l'institut Fourier

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Let $M$ be an $(n + 1)$ -dimensional Riemannian manifold admitting a covariant constant endomorphism $h$ of the localized module of 1-forms with distinct non-zero eigenvalues. After it is shown that $M$ is locally flat, a manifold $N$ immersed in $M$ is studied. The manifold $N$ has an induced structure with $n$ of the same eigenvalues if and only if the normal to $N$ is a fixed direction of $h$ . Finally conditions under which $N$ is invariant under $h$ , $N$ is totally geodesic and the induced structure has vanishing...