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  "body": "\n## Definition\n\nLet $(M_1, g_1)$ and $(M_2, g_2)$ be closed [[Riemannian manifolds|7CBT4SM83Y]].\nThey are *isospectral* if they have the same [[spectrum of the Laplace–Beltrami operator|B8TYKV7AA1]].\nThis means that each real number $\\lambda$ has the same multiplicity as an eigenvalue of the Laplacian of $M_1$ and of $M_2$.\n\n## Remarks\n\nAn [[isometry|R9YWQS15AA]] $M_1 \\to M_2$ carries eigenfunctions to eigenfunctions, so isometric manifolds are isospectral.\nBy Weyl's law, isospectral manifolds have the same dimension and the same volume.\nVariants compare the spectra of the Laplacian on $p$-forms, or the Dirichlet or Neumann spectra of domains with boundary.\n\nExample: [[Milnor's two flat tori|JNR9YJTT54]] $\\R^{16}/\\Gamma_1$ and $\\R^{16}/\\Gamma_2$, built from two non-isometric even unimodular lattices, are isospectral and not isometric.\n\nNon-example: the round spheres $S^2$ of radius $1$ and of radius $2$ are not isospectral.\nTheir eigenvalues are $k(k+1)$ and $k(k+1)/4$, so the first nonzero eigenvalues $2$ and $\\tfrac12$ differ.\n",
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