Collapse of warped submersions
Volume 89 / 2006
Abstract
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.
