Minimally separating sets, mediatrices, and Brillouin spaces
Following \textit{J. J. P. Veerman, Mauricio M. Peixoto, André C. Rocha}, and \textit{Scott Sutherland} [Commun. Math. Phys. 212, No.3, 725--744 (2000; Zbl 0979.53044)] a metric space \((X,d)\) is called a Brillouin space if it has the following properties: (i) \((X,d)\) is path-connected; (ii) \((X, d)\) is proper, i.e. for each \(y\in X\) the mapping \(d_y : X\to \mathbb{R}\) defined by \(x\mapsto d(y, x)\) is proper, i.e. it has the property that inverse images of compact sets are compact; (iii) for all \(x,y\in X\) the mediatrix \(L_{xy} = \{z\in X\mid d(x,z) = d(y,z)\}\) is minimally separating, i.e. \(X\setminus L_{xy}\) consists of more than one connected component, and no proper subset of \(L_{xy}\) has this property; (iv) \((X,d)\) is metrically consistent, i.e. for each \(x\in X\) and for every \(\varepsilon > 0\) the following property holds for all sufficiently small \(\delta > 0\): for each \(z\in X\) with \(d(x,z) = \varepsilon\) there exists a \(y\in X\) with \(d(x, y) = \delta\) such that \(U(y,d(y,z))\subset U(x,\varepsilon)\) and \(\{p\in X\mid d(x,p)=\varepsilon\}\cap \{p\in X \mid d(y,p) = d(y,z)\} = \{z\}\). In Brillouin spaces the so-called Brillouin zones, which were introduced by L. Brillouin in the 1930's and studied in solid state physics and also in focal decomposition, tile the underlying space in a very regular fashion. Therefore the main result of this paper is quite interesting. Theorem 1. Every compact, connected Riemannian manifold with the usual distance function associated with the Riemannian metric is a Brillouin space. The second important result provides a partial description of the \(\mathbb Z_2\)-homology of mediatrices in compact, connected Riemannian manifolds. Theorem 2. Let \(M\) be a compact, connected, \(n\)-dimensional Riemannian manifold, whose distance function is that associated with the Riemannian metric. Then, for any \(x,y\in M\), \(1 \leq \dim H_{n-1}(L_{xy};\mathbb Z_2)\leq\dim H_{n-1}(M;\mathbb Z_2)+1\). In a forthcoming paper it will be applied in a classification up to homeomorphism of mediatrices in some surfaces.
scientific article; zbMATH DE number 5017443
scientific article