02-193 J. J. P. Veerman

Mediatrices and Connectivity (40K, latex) Apr 22, 02

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Abstract. If $(X,d)$ is a connected metric space and $a$ and $b$ are points in $X$, then the locus $L$ of the points $x$ where $d(x,a)-d(x,b)=0$ is called a mediatrix. For instance if $X$ is a geodesic space then the geodesics emanating from $a$ and $b$ (starting at the same time, and travelling with unit speed) are said to focus at $L$. In an earlier paper (\cite{VPRS}), Brillouin Spaces were defined. These are spaces in which mediatrices have desirable properties. Most importantly: they are minimally separating. This means that for every proper subset $L'$ of $L$ in $X$, $X-L$ is disconnected, but $X-L'$ is connected. The purpose of this note is twofold. First, we give a very simple characterization of Brillouin Spaces, which shows that, for example, compact, connected Riemannian manifolds are Brillouin Spaces. Second, we give a description in terms of homology of these mediatrices. This leads to a complete topological classification of mediatrices in 2-dimensional, compact, connected, Riemannian manifolds.\\ KEYWORDS: Minimally separating, homology, geodesics, Riemannian manifolds, length spaces.

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