Continuous families of isospectral metrics on simply connected manifolds

Dorothee Schueth

Review from Zentralblatt MATH:

This paper continues interesting work done by a number of authors on constructions of continuous families of isospectral metrics on compact Riemannian manifolds, that is, isospectral deformations of Riemannian metrics, which are not isometric. Such constructions were originally obtained by forming compact quotients of a Riemannian manifold by discrete groups of isometries and began with the work of {\it C. Gordon} and {\it E. Wilson} [J. Differ. Geom. 19, No. 1, 241-256 (1984; Zbl 0922.58083)] where continuous isospectral deformations on the product $S^n\times T^m$ of the $n$-sphere for $n\geq 4$ with the $m$-dimensional torus for $m\geq 2$ are constructed so that the manifolds are not locally isometric, and deformations under which the maximum scalar curvature changes.

In this paper the author obtains similar results by considering isospectral deformations on the product $S^n\times S,$ where $S$ is a compact simply connected Lie group, in particular on $S^n\times S^3\times S^3$, and then embedding the nonsimply connected torus in $S$ and extending the metrics in such a way as to preserve the isospectrality. The nonisometry of the metrics is reflected either by different critical values of the scalar curvature function or by changes in the heat invariants for the Laplacian on 1-forms under the deformations.

Reviewed by Lew Friedland

Keywords: spectral geometry; isospectral deformation; simply connected Riemannian manifold

Classification (MSC2000): 53C20 58J53 58J50

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