Math 2.0

Global analysis, analysis on manifolds Definition

Spectral geometry

Builds on Riemannian manifoldLaplace–Beltrami operatorSpectrum of the Laplace–Beltrami operatorWeyl's lawLichnerowicz–Obata eigenvalue theorem

Definition

Let (M,g)(M, g) be a compact Riemannian manifold, possibly with boundary and a boundary condition. Let spec⁡(M,g)\operatorname{spec}(M, g) be the spectrum of the Laplace–Beltrami operator, the sequence of its eigenvalues with multiplicities. More generally, one may take the spectrum of another natural elliptic operator, such as the Hodge Laplacian on forms. Spectral geometry is the study of the relations between spec⁡(M,g)\operatorname{spec}(M, g) and the geometric invariants of (M,g)(M, g). It has two directions:

  1. direct problems: estimate or compute the eigenvalues from geometric data;
  2. inverse problems: recover geometric data from the spectrum.

Remarks

Example of a direct result: the Lichnerowicz–Obata theorem bounds the first nonzero eigenvalue below in terms of a positive lower bound on the Ricci curvature. Example of an inverse result: by Weyl's law the spectrum of a closed manifold determines its dimension and its volume. The answer to the inverse problem is not complete in general: there are non-isometric manifolds with the same spectrum.

Non-example: the study of the length of a single closed geodesic, without reference to any spectrum, is Riemannian geometry but not spectral geometry. Geodesic lengths enter spectral geometry only through the length spectrum and trace formulas.

Reading path

To understand this page from the ground up, read these 21 pages in order. The foundations come first.

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Dirichlet boundary condition, concept, no evidence levelDirichlet boundary co…conceptEigenvalue, concept, no evidence levelEigenvalueconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptLaplace–Beltrami operator, concept, no evidence levelLaplace–Beltrami oper…conceptRicci curvature, concept, no evidence levelRicci curvatureconceptRiemannian manifold, concept, no evidence levelRiemannian manifoldconceptSpectrum of the Laplace–Beltrami operator, concept, no evidence levelSpectrum of the Lapla…conceptSphere, concept, no evidence levelSphereconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptLichnerowicz–Obata eigenvalue theorem, statement, no evidence levelLichnerowicz–Obata ei…statementDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptGradient, concept, no evidence levelGradientconceptDivergence, concept, no evidence levelDivergenceconceptLaplacian, concept, no evidence levelLaplacianconceptWeyl's law, statement, no evidence levelWeyl's lawstatementSpectral geometry, definition, no evidence levelSpectral geometrydefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Dirichlet boundary condition concept
  2. Eigenvalue concept
  3. Euclidean space object
  4. Function concept
  5. Laplace–Beltrami operator concept
  6. Ricci curvature concept
  7. Riemannian manifold concept
  8. Spectrum of the Laplace–Beltrami operator concept
  9. Sphere concept
  10. Total derivative concept
  11. Function of several variables concept
  12. Vector-valued function concept
  13. Lichnerowicz–Obata eigenvalue theorem statement
  14. Differentiable function of several variables concept
  15. Partial derivative concept
  16. Vector field concept
  17. Gradient concept
  18. Divergence concept
  19. Laplacian concept
  20. Weyl's law statement
  21. Spectral geometry definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Spectral_geometry
About this page
TagB7XDAW51Z1
URIm2://B7XDAW51Z1
Kinddefinition
WritingW0 stub
MSC58J50, 58J53, 35P15
DefinesSpectral geometry
UsesRiemannian manifold, Laplace–Beltrami operator, Spectrum of the Laplace–Beltrami operator, Weyl's law, Lichnerowicz–Obata eigenvalue theorem

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