Advanced Search

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

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 $g_f{|}_{\times }\equiv g_M{|}_{\times },g_f{{|}_{\times TM̂}\equiv f̂²g_M|}_{\times TM̂}$, 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...