Mathematics > Differential Geometry
[Submitted on 21 Nov 2007]
Title:Collapsing Manifolds with Boundary
View PDFAbstract: This manuscript studies manifolds-with-boundary collapsing in the Gromov-Hausdorff topology. The main aim is an understanding of the relationship of the topology and geometry of a limiting sequence of manifolds-with-boundary to that of a limit space, which is presumed to be without geodesic terminals.
The main result establishes a disc bundle structure for any manifold-with-boundary having two-sided bounds on sectional curvature and second fundamental form, and a lower bound on intrinsic injectivity radius, which is sufficiently close in the Gromov-Hausdorff topology to a closed manifold.
The second main result identifies Gromov-Hausdorff limits of certain sequences of manifolds-with-boundary as Alexandrov spaces of curvature bounded below.
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