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Warped compact foliations

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Walczak S.

Annales Polonici Mathematici

2008

tom 94

nr 3

231-243

The notion of the Hausdorffized leaf space $\widetilde{𝓛}$ of a foliation is introduced. A sufficient condition for warped compact foliations to converge to $\widetilde{𝓛}$ is given. Moreover, a necessary condition for warped compact Hausdorff foliations to converge to $\widetilde{𝓛}$ is shown. Finally, some examples are examined.

Collapse of warped submersions

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Walczak S.

Annales Polonici Mathematici

2006

tom 89

nr 2

139-146

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.

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2 Annales Polonici Mathematici

2 Walczak S.

1 2008

1 2006

Wyniki wyszukiwania

Warped compact foliations

Collapse of warped submersions