Isospectral deformations of closed riemannian manifolds with different scalar curvature

Gordon, Carolyn S.; Gornet, Ruth; Schueth, Dorothee; Webb, David L.; Wilson, Edward N.

Annales de l'Institut Fourier, Tome 48 (1998) no. 2, pp. 593-607.

On construit les premiers exemples de familles continues de métriques riemanniennes isospectrales, mais pas localement isométriques sur des variétés fermées, plus précisément sur $S^n\times T^m$, où $T^m$ est un tore de dimension $m\ge 2$ et $S^n$ est une sphère de dimension $n\ge 4$. Ces métriques ne sont pas localement homogènes ; en particulier, la courbure scalaire d’une telle métrique n’est pas constante. Dans certaines des déformations que l’on considère, la courbure scalaire maximale change pendant la déformation.

We construct the first examples of continuous families of isospectral Riemannian metrics that are not locally isometric on closed manifolds , more precisely, on $S^n\times T^m$, where $T^m$ is a torus of dimension $m\ge 2$ and $S^n$ is a sphere of dimension $n\ge 4$. These metrics are not locally homogeneous; in particular, the scalar curvature of each metric is nonconstant. For some of the deformations, the maximum scalar curvature changes during the deformation.

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