Milnor's isospectral flat tori

Example

Let $\Gamma \subset \R^n$ be a lattice, and let $T_\Gamma = \R^n / \Gamma$ be the flat torus with the metric induced from $\R^n$. Let $\Gamma^* = \set{v \in \R^n : \ip{v}{\gamma} \in \Z \text{ for all } \gamma \in \Gamma}$ be the dual lattice. For each $\gamma^* \in \Gamma^*$, the function $x \mapsto e^{2\pi i \ip{\gamma^*}{x}}$ is an eigenfunction of the Laplacian with eigenvalue $4\pi^2 \abs{\gamma^*}^2$. These functions form an orthogonal basis of $L^2(T_\Gamma)$. So the spectrum of $T_\Gamma$ is determined by the numbers $N_{\Gamma^*}(r) = \#\set{\gamma^* \in \Gamma^* : \abs{\gamma^*}^2 = r}$.

Take $n = 16$, $\Gamma_1 = E_8 \oplus E_8$, with $E_8$ the E8 lattice, and $\Gamma_2 = D_{16}^+$. Both are even unimodular lattices, so $\Gamma_i^* = \Gamma_i$. The theta series $\sum_{\gamma \in \Gamma_i} q^{\abs{\gamma}^2/2}$ of an even unimodular lattice of rank $16$ is a modular form of weight $8$ for $SL_2(\Z)$ with constant term $1$. That space of modular forms has dimension $1$, so the two theta series are equal, and $N_{\Gamma_1}(r) = N_{\Gamma_2}(r)$ for every $r$. Hence $T_{\Gamma_1}$ and $T_{\Gamma_2}$ are isospectral.

They are not isometric. An isometry of flat tori lifts to an affine isometry of $\R^{16}$ whose linear part maps $\Gamma_1$ onto $\Gamma_2$. But the vectors of squared length $2$ generate a sublattice of index $1$ in $E_8 \oplus E_8$, and a sublattice of index $2$ in $D_{16}^+$, namely $D_{16}$.