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  "body": "\n## Summary\n\nSpectral geometry studies how the geometry of a Riemannian manifold relates to the spectra of natural differential operators on it.\nThe main such operator is the [[Laplace–Beltrami operator|F4NWZNTRMD]].\nDirect problems bound or compute eigenvalues from geometric data such as curvature, diameter and volume.\nInverse problems ask how much of the geometry is determined by the [[spectrum|B8TYKV7AA1]], as in the question of [[hearing the shape of a drum|X3VH8H1XV3]].\nIts tools include the heat kernel, [[Weyl's law|2JK9YMZV2B]] and trace formulas.\n\n## Explanations\n\n### L1 human\n\nA drum, a bell or a guitar string sounds at certain frequencies fixed by its shape.\nSpectral geometry studies this link for curved spaces: the frequencies are the eigenvalues of the Laplacian.\nFrom the frequencies one can hear the dimension, the volume and the total curvature.\nBut not everything can be heard: there exist [[isospectral manifolds|18MQEGN13R]] that sound the same and have different shapes.\n",
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