Collapse of warped submersions

Szymon M. Walczak

Annales Polonici Mathematici (2006)

  • Volume: 89, Issue: 2, page 139-146
  • ISSN: 0066-2216

Abstract

top
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 | × ≡ g M | × , g f | × T M ̂ ≡ f ̂ ² g M | × 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.

How to cite

top

Szymon M. Walczak. "Collapse of warped submersions." Annales Polonici Mathematici 89.2 (2006): 139-146. <http://eudml.org/doc/281070>.

@article{SzymonM2006,
abstract = {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 $g_f|_\{ × \} ≡ g_M|_\{ × \}, g_f|_\{× TM̂\} ≡ f̂² g_M|_\{×TM̂\}$, 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 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.},
author = {Szymon M. Walczak},
journal = {Annales Polonici Mathematici},
keywords = {Riemannian submersion; Gromov-Hausdorff topology; warped submersion},
language = {eng},
number = {2},
pages = {139-146},
title = {Collapse of warped submersions},
url = {http://eudml.org/doc/281070},
volume = {89},
year = {2006},
}

TY - JOUR
AU - Szymon M. Walczak
TI - Collapse of warped submersions
JO - Annales Polonici Mathematici
PY - 2006
VL - 89
IS - 2
SP - 139
EP - 146
AB - 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 $g_f|_{ × } ≡ g_M|_{ × }, g_f|_{× TM̂} ≡ f̂² g_M|_{×TM̂}$, 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 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.
LA - eng
KW - Riemannian submersion; Gromov-Hausdorff topology; warped submersion
UR - http://eudml.org/doc/281070
ER -

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.