Collapsing, spectrum, and Diophantine properties of Riemannian flows
type: Item
instance of: scholarly article
title: Collapsing, spectrum, and Diophantine properties of Riemannian flows
zbMATH Open document ID: 1194.58030
author: Pierre Jammes
publication date: 2010-05-05T00:00:00Z
full work available at URL: https://eudml.org/doc/116268
review text: Let \(M\) be a compact Riemannian manifold of dimension \(n\). A Riemannian flow \({\mathcal F}\) on \(M\) induces a one dimensional foliation. Consider a bundle-like metric \(g\) on \(M\), i.e., there is a splitting \(g=g_{H}+g_{V}\) of the metric \(g\) where {\parindent4mm \begin{itemize}\item[{\(\bullet\)}] \(g_{V}\) is the component of the metric tangential to the leaves; \item[{\(\bullet\)}] \(g_{H}\) is the orthogonal component; \item[{\(\bullet\)}] \(g_{H}\) is constant along the flow \({\mathcal F}\). \end{itemize}} The adiabatic collapsing associated with \(g\) is the family of metrics \(\{g_{\varepsilon}\}\) where \(g_{\varepsilon}=g_{H}+\varepsilon^{2}g_{V}\) with \(\varepsilon\leq 1\). For a non negative integer \(p\), denote, if it exists, by \(\lambda_{p,0}(M,g)\) the zero eigenvalue on \(p\)-forms, and by \[ \lambda_{p,1}(M,g)\leq\lambda_{p,2}(M,g)\leq\cdots \] the non zero eigenvalues counted with their multiplicity. On the other hand, given a vector field which is tangent to the flow, the basic cohomology \(H^{\star}(M/{\mathcal F})\) of the flow \({\mathcal F}\) is defined by the basic forms: \[ \omega\in\Omega^{\star}(M),\;\;i_{X}\omega=i_{X}d\omega=0. \] One of the main results in the paper under review may then be stated as follows: (1) the number of small non zero eigenvalues on \(p\)-forms for the collapsing \(\{g_{\varepsilon}\}\) is \[ m_{p}=\dim H^{p}(M/{\mathcal F})+\dim H^{n-p}(M/{\mathcal F})-b_{p}(M); \] (2) there exists a constant \(c>1\) such that for all \(p\), all \(1\leq k\leq m_{p}\) and all \(0<\varepsilon<1\), one has: \[ c^{-1}\varepsilon^{2}\leq\lambda_{p,k}(M,g_{\varepsilon})\leq c\varepsilon^{2}. \] Other results on the Gromov-Hausdorff distance between \((M,g_{\varepsilon})\) and the limit space of the collapsing are proved. Moreover, Diophantine properties of the flow are derived. In particular, the paper contains an appendix on arithmetic properties of Riemannian flows.
reviewed by: Salah Mehdi
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 58C40
Mathematics Subject Classification ID: 53C12
zbMATH DE Number: 5703826
zbMATH Keywords: collapsing
zbMATH Keywords: differential forms
zbMATH Keywords: Laplacian
zbMATH Keywords: small eigenvalues
zbMATH Keywords: Riemannian flows
zbMATH Keywords: Diophantine approximations
MaRDI profile type: Publication
OpenAlex ID: W2329128309
cites work: Flots riemanniens sur les 4-variétés compactes. (Riemannian flows on compact 4-manifolds)
cites work: The basic component of the mean curvature of Riemannian foliations
cites work: Adiabatic limits and spectral sequences for Riemannian foliations
cites work: Q3961493
cites work: Q3339825
cites work: Les propriétés topologiques des flots Riemanniens retrouvées à l'aide du théorème des variétés presque plates
cites work: Nilpotent Structures and Invariant Metrics on Collapsed Manifolds
cites work: A note on the first nonzero eigenvalue of the Laplacian acting on p- forms
cites work: Small eigenvalues and the Euler class of \(S^1\)-bundles
cites work: Eigenvalues of the Laplacian on Forms
cites work: La cohomologie basique d'un feuilletage riemannien est de dimension finie
cites work: Q3696054
cites work: Spectral sequences and adiabatic limits
cites work: Classification des feuilletages totalement géodésiques de codimension un
cites work: Feuilletages riemanniens sur les variétés simplement connexes
cites work: Petites valeurs propres des fibrés principaux en tores
cites work: Sur le spectre des fibrés en tore qui s'effondrent. (On the spectrum of collapsing torus bundles.)
cites work: Q5482891
cites work: Q3673702
cites work: Q5185750
cites work: Collapsing and the differential form Laplacian: the case of a smooth limit space.
cites work: Remark about the spectrum of the 𝑝-form Laplacian under a collapse with curvature bounded below
cites work: Geodesible contact structures on 3-manifolds
cites work: The adiabatic limit, Hodge cohomology and Leray's spectral sequence for a fibration
cites work: Étude des feuilletages transversalement complets et applications
cites work: Deux remarques sur les flots Riemanniens. (Two remarks on Riemannian flows)
cites work: Totally geodesic foliations, Riemannian flows and Seifert manifolds.
cites work: The Euler class for Riemannian flows
cites work: Q4532291
cites work: A finiteness theorem for foliated manifolds
cites work: Diophantine approximation
cites work: Q4337693
DOI: 10.5802/AIF.2522
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 5703826