Collapse of warped submersions

• Autorzy: Szymon M. Walczak
• Języki publikacji: 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.

Kategorie tematyczne

• 70G45: Differential-geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.)
• 53B21: Methods of Riemannian geometry

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2006
• Tom: 89
• Numer: 2
• Strony: 139-146

Daty

wydano

2006

Twórcy

autor

Szymon M. Walczak

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

Identyfikatory

DOI

10.4064/ap89-2-3