University of Illinois Urbana-Champaign

Warped products of metric spaces of curvature bounded from above

Chen, Chien-Hsiung

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https://hdl.handle.net/2142/22721

Description

Title
Warped products of metric spaces of curvature bounded from above
Author(s)
Chen, Chien-Hsiung
Issue Date
1996
Doctoral Committee Chair(s)
Bishop, Richard L.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
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.
Type of Resource
text
Permalink
http://hdl.handle.net/2142/22721
Copyright and License Information
Copyright 1996 Chen, Chien-Hsiung

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