Riemannian submersions and isoperimetric inequalities
The \(\alpha\)-isoperimetric constants, \(1\leq \alpha \leq \infty\), of a closed connected Riemannian manifold \(M^d\) are defined by \[ I_\alpha(M) = \inf_D{V_{d-1}(\partial D) \over V_d(D)^{1-1/\alpha}}, \] where \(D\) ranges over open submanifolds of \(M\) with smooth boundary such that (by reasons of symmetry) \(V_d(D) \leq {1 \over 2} V_d(M)\). Here we denote by \(V_k(N)\) the volume of a \(k\)-dimensional submanifold \(N\) with respect to the induced metric. In the paper under review, the isoperimetric constants of the total spaces of Riemannian submersions are estimated in terms of those of the basis and the fibres. The particular case of a direct product was studied in [\textit{A. A. Grigor'yan}, Mat. Zametki 38, 617-626 (1985; Zbl 0599.53035)]. A particular but representative result is the following one. Let \(\pi \colon M^d \to B^n\) be a Riemannian submersion between compact connected Riemannian manifolds. Suppose that the fibres are totally geodesic connected submanifolds of \(M\), hence they are isometric to a compact, connected Riemannian manifold \(F^k\). Then there exists a numerical constant \(C_0\) such that \[ I_\gamma(M)V_d(M)^{-1/\gamma} > C_0 \min \{ I_\alpha(F)V_k(F)^{-1/\alpha}, I_\beta(B)V_n(B)^{-1/\beta}\} \] for all \(\alpha\) and \(\beta\) with \(1 \leq \alpha, \beta \leq \infty\), where \(\gamma = \alpha + \beta\).
scientific article; zbMATH DE number 1129600