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Déformations isospectrales sur certaines nilvariétés et finitude spectrale des variétés de Heisenberg. (Isospectral deformations on certain nilmanifolds and spectral finiteness of Heisenberg manifolds). (French) Zbl 0777.58008

The author deals with isospectral deformations of left-invariant metrics \(m\) on homogeneous manifolds \(\Gamma\backslash G\), where \(G\) is a nilpotent simply connected group and \(\Gamma\subset G\) is a discrete subgroup with the factor \(\Gamma\backslash G\) compact. An automorphism \(\varphi\in\text{Aut}(G)\) is called almost inner if for every \(\lambda\in{\mathcal G}^*\) (the dual to the Lie algebra of \(G)\) there exists \(x(\lambda)\in G\) such that \(\lambda\circ\varphi_ *=\lambda(I_ x)_ *\), where \(I_ x(y)=xyx^{-1}\) is the inner automorphism. For almost inner automorphisms \(\varphi\), the measures \(m\) and \(\varphi^*m\) are isospectral.
The author asks two questions: if all inner automorphisms are inner, does there exist a nontrivial isospectral deformation, and, in general, does there exist an isospectral deformation of other kind? For so called nonsingular groups involving the case \(G={\mathcal H}_ n\) of Heisenberg groups, the answer is negative. Moreover, the author proves the rigidity of isospectral deformations, determines the classes of isometric Heisenberg manifolds \(\Gamma\backslash{\mathcal H}_ n\) and derives the finiteness of the number of classes of isometric Heisenberg manifolds that are isospectral to a given Heisenberg manifold.
Reviewer: J.Chrastina (Brno)

MSC:

58C40 Spectral theory; eigenvalue problems on manifolds
43A85 Harmonic analysis on homogeneous spaces
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References:

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