Abstract
Two compact Riemannian manifolds are said to be isospectral if the associated Laplace-Beltrami operators have the same eigenvalue spectrum. The question, often posed as “Can you hear the shape of a drum,” asks whether isospectral manifolds are necessarily isometric. The answer is now well-known to be negative; the first counterexample was a pair of isospectral tori given by Milnor in 1964. Until 1980, however, the only other examples discovered were a few additional pairs of tori or twisted products with tori.
partially supported by National Science Foundation grant DMS-8601966
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Gordon, C.S. (1992). You Can’t Hear the Shape of a Manifold. In: Tirao, J., Wallach, N.R. (eds) New Developments in Lie Theory and Their Applications. Progress in Mathematics, vol 105. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2978-0_7
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