Collapsing, spectrum, and Diophantine properties of Riemannian flows
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.
scientific article; zbMATH DE number 5703826
scientific article