Collapse of warped submersions

Szymon M. Walczak¹

¹ Faculty of Mathematics University of Łódź Banacha 22 90-238 Łódź, Poland

Annales Polonici Mathematici, Tome 89 (2006) no. 2, pp. 139-146

Résumé

We generalize the concept of warped manifold to Riemannian submersions $\pi:M\to B$ between two compact Riemannian manifolds $(M,g_M)$ and $(B,g_B)$ in the following way. If $f:B\to (0,\infty)$ is a smooth function on $B$ which is extended to a function $\widetilde f=f\circ \pi$ constant along the fibres of $\pi$ then we define a new metric $g_f$ on $M$ by $$ g_f|_{\mathcal{H}\times \mathcal{H}} \equiv g_M|_{\mathcal{H}\times \mathcal{H}},\quad\ g_f|_{\mathcal{V}\times T\widetilde M} \equiv \widetilde f^2 g_M|_{\mathcal{V}\times T\widetilde M}, $$ where $\mathcal{H}$ and $\mathcal{V}$ denote the bundles of horizontal and vertical vectors. The manifold $(M,g_f)$ obtained that way is called a warped submersion. The function $f$ is called a warping function. We show a necessary and sufficient condition for convergence of a sequence of warped submersions to the base $B$ in the Gromov–Hausdorff topology. Finally, we consider an example of a sequence of warped submersions which does not converge to its base.

Détail

DOI : 10.4064/ap89-2-3

Keywords: generalize concept warped manifold riemannian submersions between compact riemannian manifolds following infty smooth function which extended function widetilde circ constant along fibres define metric mathcal times mathcal equiv mathcal times mathcal quadg mathcal times widetilde equiv widetilde mathcal times widetilde where mathcal mathcal denote bundles horizontal vertical vectors manifold obtained called warped submersion function called warping function necessary sufficient condition convergence sequence warped submersions base gromov hausdorff topology finally consider example sequence warped submersions which does converge its base