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Isospectrality for Spherical Space Forms

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Abstract

Consider the set S(q, Γ) of isometry classes of q-dimensional spherical space forms whose fundamental groups are isomorphic to a fixed group Γ. We define a certain group \({\cal A}(q,\Gamma)\) of transformations on the finite set \({\cal S}(q,\Gamma)\), prove that any two elements in the same \({\cal A}(q,\Gamma)\)-orbit are strongly isospectral, and study some consequences. Then a number of the results are carried over to riemannian quotients of oriented real Grassmann manifolds.

Some of these results were first obtained by Ikeda, mostly for the special case (Γ cyclic) of lens spaces, and by Gilkey and Ikeda for the case where every Sylow subgroup of Γ is cyclic.

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  1. Department of Mathematics, University of California, Berkeley, California, 94720-3840, USA

    Joseph A. Wolf

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  1. Joseph A. Wolf
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Correspondence to Joseph A. Wolf.

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To my teacher, S.-S. Chern, on his 90th birthday

Research partially supported by NSF Grant DMS 99-88643.

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Wolf, J.A. Isospectrality for Spherical Space Forms. Results. Math. 40, 321–338 (2001). https://doi.org/10.1007/BF03322715

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