Strongly isospectral hyperbolic and elliptic manifolds
The author proves that if \(\Gamma_1\) and \(\Gamma_2\) are discrete fixed point free cocompact isometry groups of the hyperbolic space \(H^n\) (respectively, of elliptic space \(S^n\)), then the quotient manifolds are strongly isospectral if and only if the regular representations of the full group \(G\) of isometries of \(H^n\) (respectively \(S^n\)) into \(L^2 (\Gamma_1 \setminus G)\) and \(L^2 (\Gamma_2\setminus G)\) are equivalent. Here, strong isospectrality means isospectrality not only with respect to the Laplace operator but with respect to all natural elliptic autoadjoint operators (which include for instance the Hodge-de Rham and the Lichnerowicz operators). The result completes the work of Sunada who gave sufficient conditions for isospectrality.
scientific article; zbMATH DE number 850626