Continuous families of isospectral metrics on simply connected manifolds. (English) Zbl 0964.53027

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 C. Gordon and E. Wilson [J. Differ. Geom. 19, No. 1, 241-256 (1984; Zbl 0523.58043)]. By the nature of the construction, the examples of isospectral, nonisometric Riemannian manifolds, since they had the same Riemannian covering space, were locally isometric and had nontrivial fundamental groups so were not simply connected. Then later work of C. Gordon [J. Differ. Geom. 37, No. 3, 639-649 (1993; Zbl 0792.53037) and Contemp. Math. 173, 121-131 (1994; Zbl 0811.58063)] on closed Riemannian manifolds and recently published work of Z. Szabó [Geom. Funct. Anal. 9, No.1, 185-214 (1999; Zbl 0964.53026)] on manifolds with boundary which are isospectral but not locally isometric were followed by the results of C. Gordon and E. Wilson [J. Differ. Geom. 47, No. 3, 504-529 (1997; Zbl 0915.58104)] where continuous families of isospectral though not locally isometric Riemannian metrics were constructed on manifolds satisfying Dirichlet or Neumann conditions on the boundary. In the present paper, the author constructs continuous families of Riemannian metrics on simply connected closed manifolds which are pairwise isospectral, that is, have the same spectrum or set of eigenvalues with respect to the Laplace operator acting on 0-forms or functions, but are not locally isometric. It is interesting that the metrics constructed here are not isospectral for 1-forms since the heat invariants of the Laplacian acting on 1-forms vary with the deformations. The idea of the construction derives from the work of C. Gordon, R. Gornet, the author, D. Webb, and E. Wilson [Ann. Inst. Fourier 48, No. 2, 593-607 (1998; 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.


53C20 Global Riemannian geometry, including pinching
58J53 Isospectrality
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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