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.
For the unit and the nonnegative Laplace-Beltrami operator, the eigenvalues are , with multiplicities for . The eigenfunctions are restrictions of degree- harmonic polynomials. The harmonic decomposition of homogeneous polynomials and the Stone-Weierstrass theorem prove completeness.
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.
Suppose a finite group acts by Riemannian isometries on a compact Riemannian manifold , and Gassmann equivalent subgroups act freely. The quotient manifolds have equal Laplace-Beltrami operator spectra. For an eigenfunction space , the quotient multiplicity is , and the equality follows by summing its character of a representation over conjugacy classes. Nonconjugacy inside alone does not guarantee that the quotients are nonisometric.
For two assemblies of congruent Euclidean tiles, let encode the gluing or boundary reflection at each labelled face. An invertible constant matrix satisfying carries tile restrictions of Laplacian eigenfunctions bijectively to those on the second assembly. Boundary values satisfy and outward normal derivatives satisfy , so the intertwining identities preserve matching and boundary conditions. Use diagonal for a Dirichlet boundary condition and for a Neumann boundary condition.
A propeller domain consists of seven congruent reflected triangles: one central triangle and three arms of two triangles each. There are different coloured gluing patterns whose Dirichlet Laplacian and Neumann Laplacian spectra agree by the transplantation theorem. Varying the three side lengths of a scalene triangle in a suitable open region gives three parameters; the central triangle is identifiable from the three reflex corners, allowing a direct nonisometry proof.
A class of metrics is spectrally rigid if equality of their specified spectra forces the metrics to be isometric, or, in a deformation version, if continuous isospectral deformations are trivial. The spectrum of a flat torus determines every two-dimensional flat torus up to isometry. The Wolpert generic spectral rigidity theorem is a generic, rather than universal, uniqueness statement.
For closed hyperbolic surfaces of genus , 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.
Articles by others on the same topic
Spectral geometry is a field of mathematics that studies the relationship between the geometric properties of a manifold (a mathematical space that locally resembles Euclidean space) and the spectra of differential operators defined on that manifold, particularly the Laplace operator. Essentially, it connects the shape and structure of a geometric space to the eigenvalues and eigenfunctions of these operators.