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• # Artykuł - szczegóły

## Annales Polonici Mathematici

2006 | 89 | 2 | 139-146

## Collapse of warped submersions

EN

### Abstrakty

EN
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
$g_f|_{𝓗 × 𝓗 } ≡ g_M|_{𝓗 × 𝓗 }, g_f|_{𝓥× TM̂} ≡ f̂² g_M|_{𝓥×TM̂}$,
where 𝓗 and 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.

139-146

wydano
2006

### Twórcy

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