Collapse of warped submersions

Szymon M. Walczak

Annales Polonici Mathematici, Tome 89 (2006), p. 139-146 / Harvested from The Polish Digital Mathematics Library

Résumé

We generalize the concept of warped manifold to Riemannian submersions π: M → B between two compact Riemannian manifolds $(M, g_{M})$ and $(B, g_{B})$ in the following way. If f: B → (0,∞) is a smooth function on B which is extended to a function f̂ = f ∘ π constant along the fibres of π then we define a new metric $g_{f}$ on M by $g_{f} |_{\times} \equiv g_{M} |_{\times}, g_{f} {|_{\times T M ̂} \equiv f ̂ ² g_{M} |}_{\times T M ̂}$ , where and 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.

Publié le : 2006-01-01

Zbl 1117.53035

EUDML-ID : urn:eudml:doc:281070

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Szymon M. Walczak. Collapse of warped submersions. Annales Polonici Mathematici, Tome 89 (2006) pp. 139-146. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap89-2-3/