Brillouin zones in spaces of constant curvature

In this section X will be assumed to be one of $\mbox{$\Bbb R$ }^n$, $\mbox{$\Bbb S$ }^n$, or $\mbox{$\Bbb H$ }^n$, all equipped with the standard metric, and let G be a discontinuous group of isometries of X. Denote the quotient X/G with the induced metric by (M,g). Then the construction of lifting to the universal cover, as outlined in the introduction, applies naturally to (M,g). In this section we describe focusing of geodesics in (M,g) by Brillouin zones in X. The discrete set S is given by the orbit of a chosen point in X (which we will call the origin) under the group of deck-transformations G. The fact that the Brillouin zones are fundamental domains is now a direct corollary of proposition 2.11.

The regularity conditions of Def. 2.5 are easily verified in the present context. We do this first.

Lemma 3.1   If X is either $\mbox{$\Bbb R$ }^n$, $\mbox{$\Bbb S$ }^n$, or $\mbox{$\Bbb H$ }^n$, then a mediatrix L_ab in X is an (n-1)-dimensional, totally geodesic subspace consisting of one component, and X-L_ab has two components.

Proof: This is easy to see if we change coordinates by an isometry of X, putting a and b in a convenient position, say as x and -x. The mediatrix L_x,-x is easily seen to satisfy the conditions (in the case of $\mbox{$\Bbb S$ }^n$, it is the equator, and for the others, it is a hyperplane). The conclusion follows. $\Box$