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  "body": "\n## Definition\n\nLet $(M, g)$ be a compact [[Riemannian manifold|7CBT4SM83Y]], possibly with boundary and a boundary condition.\nLet $\\operatorname{spec}(M, g)$ be the [[spectrum of the Laplace–Beltrami operator|B8TYKV7AA1]], the sequence of its eigenvalues with multiplicities.\nMore generally, one may take the spectrum of another natural elliptic operator, such as the Hodge Laplacian on forms.\n*Spectral geometry* is the study of the relations between $\\operatorname{spec}(M, g)$ and the geometric invariants of $(M, g)$.\nIt has two directions:\n\n1. direct problems: estimate or compute the eigenvalues from geometric data;\n2. inverse problems: recover geometric data from the spectrum.\n\n## Remarks\n\nExample of a direct result: the [[Lichnerowicz–Obata theorem|TK1PT9FDV0]] bounds the first nonzero eigenvalue below in terms of a positive lower bound on the Ricci curvature.\nExample of an inverse result: by [[Weyl's law|2JK9YMZV2B]] the spectrum of a closed manifold determines its dimension and its volume.\nThe answer to the inverse problem is not complete in general: there are non-isometric manifolds with the same spectrum.\n\nNon-example: the study of the length of a single closed geodesic, without reference to any spectrum, is Riemannian geometry but not spectral geometry.\nGeodesic lengths enter spectral geometry only through the length spectrum and trace formulas.\n",
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