Spectral geometry is a field in

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

which concerns relationships between geometric structures of

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

s and spectra of canonically defined

differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...

s. The case of the

Laplace–Beltrami operator In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named af ...

on a

closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...

Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...

has been most intensively studied, although other

Laplace operators in differential geometry In differential geometry there are a number of second-order, linear, elliptic differential operators bearing the name Laplacian. This article provides an overview of some of them. Connection Laplacian The connection Laplacian, also known as the ...

have also been examined. The field concerns itself with two kinds of questions: direct problems and inverse problems. Inverse problems seek to identify features of the geometry from information about the eigenvalues of the Laplacian. One of the earliest results of this kind was due to

Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...

who used

David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...

's theory of integral equation in 1911 to show that the volume of a bounded domain in Euclidean space can be determined from the asymptotic behavior of the eigenvalues for the Dirichlet boundary value problem of the

Laplace operator In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...

. This question is usually expressed as " Can one hear the shape of a drum?", the popular phrase due to Mark Kac. A refinement of Weyl's asymptotic formula obtained by Pleijel and Minakshisundaram produces a series of local

spectral invariants In symplectic geometry, the spectral invariants are invariants defined for the group of Hamiltonian diffeomorphisms of a symplectic manifold, which is closed related to Floer theory and Hofer geometry. Arnold conjecture and Hamiltonian Floer homo ...

involving covariant differentiations of the curvature tensor, which can be used to establish spectral rigidity for a special class of manifolds. However as the example given by John Milnor tells us, the information of eigenvalues is not enough to determine the isometry class of a manifold (see isospectral). A general and systematic method due to Toshikazu Sunada gave rise to a veritable cottage industry of such examples which clarifies the phenomenon of isospectral manifolds. Direct problems attempt to infer the behavior of the eigenvalues of a Riemannian manifold from knowledge of the geometry. The solutions to direct problems are typified by the Cheeger inequality which gives a relation between the first positive eigenvalue and an isoperimetric constant (the Cheeger constant). Many versions of the inequality have been established since Cheeger's work (by R. Brooks and P. Buser for instance).

References

* . * . {{SpectralTheory Differential geometry Spectral theory Riemannian geometry

See also

References