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Quantizing a moduli-space approximation gives wavefunctions with measure $\sqrt{\det g}{d}^{d}q$ and, under the minimal scalar ordering, the Hamiltonian operator $-{\hslash }^2{\Delta }_g/2+V$, where ${\Delta }_g$ is the Laplace-Beltrami operator. Global identifications, statistics and regularity supply additional restrictions on wavefunctions. Curvature-dependent ordering terms or quantum corrections are extra choices, not determined by the classical kinetic energy alone.

For a geodesic $\gamma$ with initial velocity $v$, $(f\circ \gamma {)}^{\prime \prime }(0)=\mathrm{Hess}f(v,v)$, because the covariant acceleration vanishes. Summing over an orthonormal basis of the tangent space gives $\Delta f(p)=-{\sum }_i(f\circ {\gamma }_i{)}^{\prime \prime }(0)$ for the nonnegative Laplace-Beltrami operator.

Two Riemannian manifolds are isospectral for a specified differential operator if their eigenvalues, counted with multiplicities and with the same boundary conditions, agree. Riemannian isometries preserve the Laplace-Beltrami operator spectrum, but the converse can fail. The transplantation theorem and the Sunada theorem give systematic constructions.

If a twice differentiable real function has a local maximum at an interior point of a Riemannian manifold, its Hessian matrix is negative semidefinite there and hence its Laplacian is nonpositive.

The unmarked length spectrum of a compact hyperbolic surface is the multiset of lengths of its closed geodesics, with a consistent convention for primitive curves and orientation. The marked version labels lengths by free homotopy classes. The Selberg trace formula relates the unmarked multiset to the Laplace-Beltrami operator spectrum.