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The spectral characterization of Riemannian manifolds is pretty convincing, no? Seea good example in my opinion. Details are given in the following article: http://www.alainconnes.org/docs/ckm1.pdf .

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The spectral characterization of Riemannian manifolds is pretty convincing, no? See: http://www.alainconnes.org/docs/ckm1.pdf .