The topology of surface mediatrices

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Topology and its Applications, ISSN: 0166-8641, Vol: 154, Issue: 1, Page: 54-68

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Bernhard, James; Veerman, J. J. P.
Elsevier BV
Mathematics; minimally separating sets; geodesics; compact surfaces; simplicial complexes; mathematics; Simplicial complexes; Geodesics (Mathematics); Topology; Manifolds (Mathematics); Riemann surfaces; Mathematics and Computer Science; Geometry and Topology
article description
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.