Math 2.0

Global analysis, analysis on manifolds Index theory Example

Index of the de Rham complex

Builds on Analytic indexTopological indexHodge theoremEuler characteristicChern–Gauss–Bonnet theoremBetti number

Example

Let MM be a closed oriented Riemannian manifold of dimension nn, and let Ωk(M)\Omega^k(M) be the space of smooth real kk-forms. Let d∗d^* be the formal adjoint of the exterior derivative dd. The operator D=d+d∗ ⁣:Ωeven(M)→Ωodd(M)D = d + d^* \colon \Omega^{\mathrm{even}}(M) \to \Omega^{\mathrm{odd}}(M) is elliptic, since D∗DD^* D is the Hodge Laplacian on even forms. Its kernel is the space of harmonic even forms, and its cokernel is isomorphic to ker⁡D∗\ker D^*, the space of harmonic odd forms. By the Hodge theorem, the harmonic kk-forms form a space of dimension bkb_k, the kk-th Betti number. So the analytic index is ind⁡D=∑k=0n(−1)kbk=χ(M),\operatorname{ind} D = \sum_{k = 0}^{n} (-1)^k b_k = \chi(M), the Euler characteristic of MM.

Remarks

For nn odd, χ(M)=0\chi(M) = 0 by Poincaré duality, and the index is 00. For nn even, the topological index of DD is the Euler number ⟨e(TM),[M]⟩\langle e(TM), [M] \rangle. So the Atiyah–Singer index theorem gives χ(M)=⟨e(TM),[M]⟩\chi(M) = \langle e(TM), [M] \rangle, and with the Chern–Weil form of the Euler class this is the Chern–Gauss–Bonnet theorem. For the round sphere S2S^2: b0=b2=1b_0 = b_2 = 1 and b1=0b_1 = 0, so the index is 22.

Reading path

To understand this page from the ground up, read these 23 pages in order. The foundations come first.

Show the path as a graph
Analytic index, concept, no evidence levelAnalytic indexconceptBetti number, concept, no evidence levelBetti numberconceptCurvature form, concept, no evidence levelCurvature formconceptDifferential form, concept, no evidence levelDifferential formconceptEuler characteristic, concept, no evidence levelEuler characteristicconceptGaussian curvature, concept, no evidence levelGaussian curvatureconceptGeodesic curvature, concept, no evidence levelGeodesic curvatureconceptHodge star operator, concept, no evidence levelHodge star operatorconceptLevi-Civita connection, concept, no evidence levelLevi-Civita connectionconceptOrientation of a manifold, concept, no evidence levelOrientation of a mani…conceptParametrised surface, concept, no evidence levelParametrised surfaceconceptPfaffian, concept, no evidence levelPfaffianconceptRegular surface, concept, no evidence levelRegular surfaceconceptRiemannian manifold, concept, no evidence levelRiemannian manifoldconceptTopological index, concept, no evidence levelTopological indexconceptde Rham cohomology, concept, no evidence levelde Rham cohomologyconceptExterior derivative, concept, no evidence levelExterior derivativeconceptTangent plane and normal vector of a surface, concept, no evidence levelTangent plane and nor…conceptOrientable surface, concept, no evidence levelOrientable surfaceconceptHodge theorem, statement, no evidence levelHodge theoremstatementGauss–Bonnet theorem, statement, no evidence levelGauss–Bonnet theoremstatementChern–Gauss–Bonnet theorem, statement, no evidence levelChern–Gauss–Bonnet th…statementIndex of the de Rham complex, example, no evidence levelIndex of the de Rham …example
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. Analytic index concept
  2. Betti number concept
  3. Curvature form concept
  4. Differential form concept
  5. Euler characteristic concept
  6. Gaussian curvature concept
  7. Geodesic curvature concept
  8. Hodge star operator concept
  9. Levi-Civita connection concept
  10. Orientation of a manifold concept
  11. Parametrised surface concept
  12. Pfaffian concept
  13. Regular surface concept
  14. Riemannian manifold concept
  15. Topological index concept
  16. de Rham cohomology concept
  17. Exterior derivative concept
  18. Tangent plane and normal vector of a surface concept
  19. Orientable surface concept
  20. Hodge theorem statement
  21. Gauss–Bonnet theorem statement
  22. Chern–Gauss–Bonnet theorem statement
  23. Index of the de Rham complex example

Further reading

  1. Web https://en.wikipedia.org/wiki/Atiyah–Singer_index_theorem
About this page
Tag2DYR4C5Q5V
URIm2://2DYR4C5Q5V
Kindexample
WritingW0 stub
MSC58J20, 58A12
IllustratesAnalytic index, Topological index
UsesAnalytic index, Topological index, Hodge theorem, Euler characteristic, Chern–Gauss–Bonnet theorem, Betti number

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