Lichnerowicz FormulaEdit
The Lichnerowicz formula is one of the central results in spin geometry and global analysis. It provides a precise link between analysis on spinor fields and the curvature of the underlying manifold. Concretely, it expresses the square of the Dirac operator in terms of a connection Laplacian on spinors and a curvature term, revealing how scalar curvature controls spinor fields and leading to powerful vanishing theorems and topological consequences.
In its closest form, the formula appears on a compact spin manifold and ties together the Dirac operator, the spin connection, and the scalar curvature. This bridge between analysis and geometry is a cornerstone of modern techniques in differential geometry and global analysis, with important repercussions in index theory and mathematical physics.
Lichnerowicz formula
Let (M,g) be a spin manifold of dimension n with spinor bundle ΣM. Denote by D the Dirac operator acting on sections of ΣM, and by ∇ the spin connection on ΣM (the lift of the Levi-Civita connection). The scalar curvature of M is Scal. Then for every spinor field ψ ∈ Γ(ΣM) the Lichnerowicz formula states
D^2 ψ = ∇^* ∇ ψ + (Scal/4) ψ,
where ∇^* ∇ is the rough Laplacian (the connection Laplacian) on spinors. Equivalently, in a local orthonormal frame {e_i},
D = ∑i e_i · ∇{e_i}^S, and D^2 = ∇^* ∇ + (Scal/4),
with the Clifford multiplication by e_i denoted by · and ∇^S the spin connection.
This identity is a specific instance of the more general Weitzenböck formula, which relates square of Dirac-type operators to Laplacians plus curvature terms. The Lichnerowicz formula shows how the scalar curvature directly influences the behavior of spinor fields.
Consequences of the formula are immediate once one imposes curvature conditions. If M is compact and Scal > 0 everywhere, then any harmonic spinor (Dψ = 0) must vanish, because the Lichnerowicz identity implies ∫( |∇ψ|^2 + (Scal/4)|ψ|^2 ) = 0, forcing ψ to be identically zero. This simple observation has far-reaching topological implications when combined with index theory.
The formula also yields eigenvalue estimates for D. Friedrich’s inequality provides a lower bound for nonzero eigenvalues λ of D in terms of the minimum of Scal:
λ^2 ≥ (n/4(n−1)) · inf_M Scal.
Such estimates connect the geometry of M to the spectral properties of D, with further consequences in geometric analysis and mathematical physics.
The Lichnerowicz identity is especially powerful when viewed through the lens of the Atiyah-Singer index theorem. For a compact spin manifold, the index of the positive part of D is equal to the Â-genus of M. The vanishing results coming from D^2 and Scal thus translate into topological obstructions: manifolds with nonzero Â-genus cannot admit metrics of everywhere positive scalar curvature. This interplay between analysis, topology, and geometry is a central theme in the study of manifolds and their curvature.
Consequences, applications, and generalizations
vanishing theorems: On compact manifolds with positive scalar curvature, the Dirac operator has trivial kernel. This yields restrictions on the topology of the manifold via the index theorem (e.g., the Â-genus must vanish in certain cases).
topology and obstructions to positive curvature: The Lichnerowicz formula provides analytic obstructions to the existence of metrics of positive scalar curvature on spin manifolds. The interplay with index theory makes precise statements about when such metrics can exist, feeding into broader programs in differential topology.
eigenvalue estimates: The inequality for Dirac eigenvalues ties curvature to spectral data, informing questions in spectral geometry and mathematical physics.
generalizations and twisted settings: The formula extends to twisted Dirac operators, where the Dirac operator acts on sections of S ⊗ E for a vector bundle E with connection. The Weitzenböck-type identity acquires additional curvature terms from the twisting connection, enriching the interplay between curvature, topology, and analysis. In more general settings, analogous formulas appear for spin^c manifolds and other Dirac-type operators, maintaining the same spirit of connecting curvature to analytical behavior.
physical relevance: In curved spacetime physics, Dirac operators model fermionic fields, and the Lichnerowicz-type identities inform energy estimates, stability considerations, and aspects of quantum field theory in curved backgrounds. They also connect to global results in mathematical physics, such as the use of spinors in the positive mass theorem and related geometric-analytic methods.
related theorems and notions: The Lichnerowicz formula sits alongside a family of powerful tools in differential geometry, including the Weitzenböck formula, spin geometry, and index theory, all of which illuminate how local curvature affects global analytic and topological properties.