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The Topology of Surface Mediatrices
Mathematics and Statistics Faculty Publications and Presentations
  • James Bernhard, University of Puget Sound
  • J. J. P. Veerman, Portland State University
Document Type
Publication Date
  • Simplicial complexes,
  • Geodesics (Mathematics),
  • Topology,
  • Manifolds (Mathematics),
  • Riemann surfaces
Given a pair of distinct points p and q in a metric space with distance d, the mediatrix is the set of points x such that d(x,p)=d(x,q). In this paper, we examine the topological structure of mediatrices in connected, compact, closed 2-manifolds whose distance function is inherited from a Riemannian metric. We determine that such mediatrices are, up to homeomorphism, finite, closed simplicial 1-complexes with an even number of incipient edges emanating from each vertex. Using this and results from [J.J.P. Veerman, J. Bernhard, Minimally separating sets, mediatrices and Brillouin spaces, Topology Appl., in press], we give the classification up to homeomorphism of mediatrices on genus 1 tori (and on projective planes) and outline a method which may possibly be used to classify mediatrices on higher-genus surfaces.

This is the author’s version of a work that was accepted for publication in Topology and its Applications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication.

A definitive version was subsequently published in Topology and its Applications and can be found online at:

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Citation Information
James Bernhard and J. J. P. Veerman. "The Topology of Surface Mediatrices" (2007)
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