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Spectral geometry studies which properties of a Riemannian manifold can be recovered from the spectrum of geometric differential operators, especially the Laplace-Beltrami operator. Explicit spherical harmonics, Fourier series on a flat torus, and isospectral manifolds illustrate three different aspects of the problem.

With the nonnegative Laplace-Beltrami operator, a frequency $w$ in the dual lattice gives the eigenfunction $e^{2\pi i⟨w,x⟩}$ and eigenvalue $4{\pi }^2|w{|}^2$. Its multiplicity is the number of dual vectors with that norm. Fourier series prove completeness. In dimension two the shortest vector, shortest independent vector and covolume determine a reduced Gram matrix, proving spectral rigidity.

For the unit $S^d$ and the nonnegative Laplace-Beltrami operator, the eigenvalues are $\ell (\ell +d-1)$, with multiplicities $\left(\genfrac{}{}{0ex}{}{\ell +d}{d}\right)-\left(\genfrac{}{}{0ex}{}{\ell +d-2}{d}\right)$ for $\ell \ge 0$. The eigenfunctions are restrictions of degree-$\ell$ harmonic polynomials. The harmonic decomposition of homogeneous polynomials and the Stone-Weierstrass theorem prove completeness.

For the positive Laplace-Beltrami operator on the unit sphere $S^d$, with $d\ge 1$, the eigenvalues are $n(n+d-1)$ for integers $n\ge 0$. Their eigenspaces consist of restrictions of degree-$n$ harmonic polynomials and have dimensions $\left(\genfrac{}{}{0ex}{}{n+d}{d}\right)-\left(\genfrac{}{}{0ex}{}{n+d-2}{d}\right)$, with the second term zero for $n<2$. The Laplacian in polar coordinates gives the eigenvalue; Fischer decomposition and the Stone-Weierstrass theorem prove completeness.

For a smooth real function on the unit two-sphere, integration of the covariant-derivative commutation formula gives the displayed identity. Explicitly, ${\nabla }^a{\nabla }_a{\nabla }_{b}w={\nabla }_b{\Delta }_{S}w+{\mathrm{Ric}}_b{}^c{\nabla }_{c}w$, and the unit sphere has $\mathrm{Ric}=g$. Integrating by parts once and then again proves the formula. It controls all angular second derivatives by the Laplace-Beltrami operator without singular estimates at coordinate poles.

Suppose a finite group $G$ acts by Riemannian isometries on a compact Riemannian manifold $M$, and Gassmann equivalent subgroups $H_1,H_2$ act freely. The quotient manifolds $H_i\backslash M$ have equal Laplace-Beltrami operator spectra. For an eigenfunction space $E$, the quotient multiplicity is $|H_i{|}^{-1}{\sum }_{h\in H_i}\mathrm{tr}(h|E)$, and the equality follows by summing its character of a representation over conjugacy classes. Nonconjugacy inside $G$ alone does not guarantee that the quotients are nonisometric.

The surface Laplacian is the Laplace-Beltrami operator of a surface with its induced Riemannian metric. It is ${\Delta }_{s}f={\nabla }_s\cdot {\nabla }_{s}f$, using the surface gradient and surface divergence. On a sphere of radius $a$, a degree-$l$ spherical harmonic has eigenvalue $-l(l+1)/a^2$.

A pair of closed orientable flat three-manifolds with equal scalar Laplace-Beltrami operator spectra but different first Betti numbers, respectively one and zero. Consequently they are not homeomorphic. They cannot be isospectral for the Hodge Laplacian on one-forms, whose zero-eigenvalue multiplicity is the first Betti number. This shows why the spectrum on functions must not be silently replaced by the spectra on all differential forms.

For the positive scalar Laplace-Beltrami operator on a closed $d$-dimensional Riemannian manifold, the number of eigenvalues at most $\lambda$, counted with multiplicity, has the displayed leading asymptotic. The exponent determines the dimension and the coefficient determines the total volume of the Riemannian metric. There is an analogous result with fixed Dirichlet boundary conditions or Neumann boundary conditions on a compact smooth domain.

For closed hyperbolic surfaces of genus $g\ge 2$, there is a closed proper real-analytic exceptional subset of Teichmüller space such that a surface outside it is determined up to isometry by its unmarked length spectrum, equivalently its Laplace-Beltrami operator spectrum. The analytic part of the proof controls possible spectral matchings using finite determining length data; the geometric part recognizes persistent matchings as changes of marking, using the collar lemma and variations of Fenchel–Nielsen coordinates. Orientation reversal remains invisible to the spectrum.