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Riemannian manifolds isospectral on functions but not on 1-forms. (English) Zbl 0585.53036
For (M,G) a compact Riemannian manifold, let $$spec^ p (M,g)$$ denote the spectrum of the Laplace-Beltrami operator acting on the space of smooth p-forms on M. Two manifolds (M,g) and (M’,g’) will be said to be p- isospectral if $$spec^ p (M,g)=spec^ p (M',g')$$. It would be of interest to determine whether for each k, the collection of all $$spec^ p (M,g)$$, $$p=0,1,...,k$$ contains more information than does $$spec^ p (M,g)$$, $$p=0,1,...,k-1$$. We answer the question affirmatively when $$k=1$$ by constructing examples of 0-isospectral manifolds which are not 1- isospectral. The manifolds involved are compact quotients of Heisenberg groups.

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