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Warped products of metric spaces of curvature bounded from above

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Title: Warped products of metric spaces of curvature bounded from above
Author(s): Chen, Chien-Hsiung
Doctoral Committee Chair(s): Bishop, R. L.
Department / Program: Mathematics
Discipline: Mathematics
Degree Granting Institution: University of Illinois at Urbana-Champaign
Degree: Ph.D.
Genre: Dissertation
Subject(s): Mathematics
Abstract: In this work we extend the idea of warped products, which was previously defined on smooth Riemannian manifolds, to geodesic metric spaces and prove the analogue of the theorems on spaces with curvature bounded from above.Suppose that function $f\ :\ M\to R\sp{+}$ is continuous and ($M\times\sb{f}\ N,\ d)$ denotes the warped product of two metric spaces $(M,\ d\sb{M})$ and $(N,\ d\sb{N})$. We prove the following main results in this thesis.Theorem. If $(M,\ d\sb{M})$ and $(N,\ d\sb{N})$ are geodesic metric spaces and if $(M\times\sb{f}\ N,\ d)$ has nonpositive curvature, then (1) $(M,\ d\sb{M})$ has nonpositive curvature. (2) $(N,\ d\sb{N})$ has nonpositive curvature if f has a minimum. (3) f is convex.Theorem. Let M be R or a graph. If $(N,\ d\sb{N})$ has nonpositive curvature and $f\ :\ M\to R\sp{+}$ is convex then $(M\times\sb{f}\ N,\ d)$ has nonpositive curvature.
Issue Date: 1996
Type: Text
Language: English
URI: http://hdl.handle.net/2142/22721
ISBN: 9780591197631
Rights Information: Copyright 1996 Chen, Chien-Hsiung
Date Available in IDEALS: 2011-05-07
Identifier in Online Catalog: AAI9712222
OCLC Identifier: (UMI)AAI9712222
 

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