Lichnerowicz TheoremEdit
The Lichnerowicz Theorem sits at a crossroads of differential geometry, global analysis, and topology. It ties the curvature of a space to the behavior of spinor fields through the Dirac operator, exposing deep constraints on how a space can be curved if certain spinorial structures are present. Named for the French-Polish mathematician André Lichnerowicz, the result is a cornerstone in spin geometry and has influenced developments ranging from index theory to questions about which spaces can admit metrics of positive scalar curvature.
At the heart of the theorem is a precise identity that expresses the square of the Dirac operator in terms of a rough Laplacian on spinors plus a curvature term. This identity, together with a compactness or spectral argument, yields vanishing results for harmonic spinors under positive curvature conditions and connects geometric properties to topological invariants. The theorem is often discussed alongside the broader framework of the Weitzenböck formulas and the Atiyah–Singer index theorem, which together illuminate how analysis and topology constrain each other on manifolds with extra geometric structure.
Lichnerowicz Theorem
Statement
Let M be a compact spin manifold equipped with a Riemannian metric g. Let D denote the Dirac operator acting on smooth sections of the spinor bundle ΣM, and let s be the scalar curvature of g. If s is positive everywhere on M, then the kernel of D is trivial; that is, there are no nonzero harmonic spinors. Consequently, the index of the chiral Dirac operator D^+ vanishes, which implies that certain topological invariants, such as the Â-genus of M, must be zero in this setting.
The Lichnerowicz formula
A key ingredient is the Lichnerowicz (or Weitzenböck) formula: D^2 = ∇^∇ + (s/4), where ∇ is the spin connection on ΣM and ∇^∇ is the rough Laplacian on spinors. This identity translates curvature information into a spectral statement about D. In particular, if Dψ = 0 for a spinor ψ, then the formula yields 0 = ⟨D^2ψ, ψ⟩ = ⟨∇ψ, ∇ψ⟩ + (s/4)|ψ|^2, and because s > 0, the only solution is ψ ≡ 0.
Consequences for scalar curvature and topology
- Vanishing of harmonic spinors: On a closed spin manifold with s > 0, ker(D) = {0}. This is the central vanishing statement of the theorem.
- Topological obstructions to positive curvature: The index of the Dirac operator, which equals the Â-genus Â(M), must vanish if s > 0. Therefore, manifolds with nonzero Â-genus cannot carry metrics of everywhere positive scalar curvature.
- Nontrivial spin-topology constraints: The interplay between the Dirac operator and Â(M) links curvature to global topological data, showing that geometry cannot be freely chosen without regard to the manifold’s spin and topological structure.
Examples and special cases
- K3 surfaces and many other spin 4-manifolds with nonzero Â-genus cannot admit metrics of positive scalar curvature, reflecting the index-theoretic obstruction provided by the theorem.
- In contrast, certain manifolds with zero Â-genus or with appropriate geometric features can support metrics of positive scalar curvature, illustrating the sharpness of the obstruction in the presence of spin structures.
Extensions and related results
- Friedrich inequality: On a closed spin manifold of dimension n, eigenvalues λ of the Dirac operator satisfy a lower bound λ^2 ≥ (n/4(n−1)) inf s(x). This gives quantitative control over the spectrum in terms of scalar curvature.
- General Weitzenböck framework: The D^2 formula generalizes to other natural differential operators, yielding a family of vanishing or spectral estimates tied to curvature.
- Connections to index theory: The Lichnerowicz formula provides a concrete analytic manifestation of the Atiyah–Singer index theorem, linking spectral data of D to topological invariants like the Â-genus.
Historical context
The result is attributed to André Lichnerowicz, whose work in the 1960s developed the spinorial toolkit that connects Dirac operators to curvature. The formula D^2 = ∇^*∇ + (s/4) and its consequences were early instances where a local curvature measure dictated global analytic behavior, foreshadowing later triumphs in index theory and global analysis. The ideas have since been extended and refined, becoming standard tools in the study of spin geometry, the geometry of Dirac operators, and the geometry–topology interface.