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Zbl 0777.58008
Pesce, Hubert
Déformations isospectrales sur certaines nilvariétés et finitude spectrale des variétés de Heisenberg. (Isospectral deformations on certain nilmanifolds and spectral finiteness of Heisenberg manifolds). (French)
[J] Ann. Sci. Éc. Norm. Supér. (4) 25, No. 5, 515-538 (1992). ISSN 0012-9593

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 $\lambda\in{\cal G}\sp*$ (the dual to the Lie algebra of $G)$ there exists $x(\lambda)\in G$ such that $\lambda\circ\varphi\sb *=\lambda(I\sb x)\sb *$, where $I\sb x(y)=xyx\sp{-1}$ is the inner automorphism. For almost inner automorphisms $\varphi$, the measures $m$ and $\varphi\sp*m$ are isospectral.\par 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 $G={\cal H}\sb n$ of Heisenberg groups, the answer is negative. Moreover, the author proves the rigidity of isospectral deformations, determines the classes of isometric Heisenberg manifolds $\Gamma\backslash{\cal H}\sb n$ and derives the finiteness of the number of classes of isometric Heisenberg manifolds that are isospectral to a given Heisenberg manifold.
[J.Chrastina (Brno)]

MSC 2000:: *58C40 Spectral theory on manifolds
43A85 Analysis on homogeneous spaces

Keywords: nilpotent Lie group; isospectral deformations; almost linear automorphisms

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