Locally non-isometric yet super isospectral spaces
The Laplace-Beltrami operator acting on functions on a closed Riemannian manifold has discrete real spectrum. It was a famous question whether or not this spectrum determines the manifold up to isometry. The question was answered to the negative by \textit{J. W. Milnor} [Proc. Nat. Acad. Sci. USA 51, 542 (1964; Zbl 0124.31202)] and since then many mathematicians have constructed pairs of isospectral non-isometric Riemannian manifolds and even families of such (e.g. \textit{M.-F. Vignéras} [Ann. Math., II. Ser. 112, 21-32 (1980; Zbl 0445.53026)], \textit{C. Gordon} and \textit{E. Wilson} [J. Differ. Geom. 19, 241-256 (1984; Zbl 0523.58043)], \textit{T. Sunada} [Ann. Math., II. Ser. 121, 169-186 (1985; Zbl 0585.58047)]). All these examples are locally isometric. The first locally non-isometric examples were published by \textit{C. Gordon} [J. Differ. Geom. 37, No. 3, 639-649 (1993; Zbl 0792.53037)]. In the present paper the author constructs further pairs of locally non-isometric but isospectral compact Riemannian manifolds with and without boundary. The construction goes as follows: Let \(a\) and \(b\) be natural numbers, let \({\mathbb H} \cong {\mathbb R}^4\) denote the quaternions. If we identify \(\mathbb R^3\) with the imaginary quaternions, then \({\mathbb R}^3\) acts on \({\mathbb H}^a\) by quaternionic multiplication from the right and on \({\mathbb H}^b\) from the left. From this one constructs a Heisenberg type Lie algebra structure on \({\mathbb H}^a \times {\mathbb H}^b \times {\mathbb R}^3\) and hence by exponentiating a Lie group structure. By pushing forward a natural invariant metric to the quotient of \({\mathbb H}^a \times {\mathbb H}^b \times {\mathbb R}^3\) by the subgroup \(\{0\} \times \{0\} \times {\mathbb Z}^3\) one obtains a Riemannian manifold \({\mathbb CH}^{(a,b)}_3\) diffeomorphic to \({\mathbb R}^{4(a+b)}\times T^3\). Let \(B\subset {\mathbb R}^{4(a+b)}\) be a ball about the origin and let \(S\) be its boundary. It turns out that for different pairs \((a,b)\) with the same \(a+b\) the manifolds \(B\times T^3 \subset {\mathbb CH}^{(a,b)}_3\) have the same Laplace spectrum (with both Dirichlet and Neumann boundary conditions) also the closed manifolds \(S\times T^3 \subset {\mathbb CH}^{(a,b)}_3\) are isospectral. This is proved by constructing operators intertwining the Laplace operators. These mainfolds are not locally isometric unless the corresponding \(a\)'s and \(b\)'s coincide up to order. The isometry groups of these isospectral manifolds even have different dimensions. Different examples of non-isometric yet isospectral metrics (even families of such) on products of spheres and tori have been constructed by \textit{C. Gordon, R. Gornet, D. Schueth, D. Webb}, and \textit{E. Wilson} [Ann. Inst. Fourier 48, No. 2, 593-607 (1998; Zbl 0922.58083)]. The first simply-connected examples have been found by \textit{D. Schueth} [Ann. Math. (2) 149, No. 1, 287-308 (1999; Zbl 0964.53027), see the review below].
scientific article; zbMATH DE number 1286130