Lichnerowicz formula
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| Lichnerowicz formula | |
|---|---|
| Name | Lichnerowicz formula |
| Field | Differential geometry |
| Introduced | 1960s |
| Introduced by | André Lichnerowicz |
| Related | Dirac operator, Atiyah–Singer index theorem, Weitzenböck identity |
Lichnerowicz formula
The Lichnerowicz formula relates the square of a Dirac-type operator on a spin manifold to a Laplace-type operator plus scalar curvature terms, connecting analysis, topology, and physics. It plays a central role in studies influenced by André Lichnerowicz, Claude Chevalley, Michael Atiyah, Isadore Singer, and Edward Witten, and informs work in global analysis on manifolds studied by Shing-Tung Yau, Mikhail Gromov, and Henri Cartan.
Introduction
The formula arises in the context of spin geometry developed by Élie Cartan, Wilhelm Killing, Hermann Weyl, and Rudolf Lipschitz, and it became prominent through interactions among André Lichnerowicz, Atiyah, Singer, and Roger Penrose. It connects operators studied by Paul Dirac, Hermann Weyl, and David Hilbert with curvature notions examined by Bernhard Riemann, Gregorio Ricci-Curbastro, and Tullio Levi-Civita, and it influenced later developments by Simon Donaldson, Karen Uhlenbeck, and Atiyah–Patodi–Singer.
Statement of the formula
On a compact Riemannian spin manifold studied by Lichnerowicz and Narasimhan, with spinor bundle built from representations similar to those used by Hermann Weyl and Élie Cartan, the formula expresses D^2 in terms of the Bochner Laplacian associated to Levi-Civita connections of Riemann and Ricci curvature tensors introduced by Riemann and Ricci. The classical statement used by Atiyah and Singer reads: D^2 = ∇^*∇ + (1/4)R, where D is the Dirac operator formulated in the tradition of Paul Dirac and Hermann Weyl, ∇ is the spin connection related to Levi-Civita, and R is the scalar curvature appearing in Riemann's work and later studied by Marcel Berger, Shiing-Shen Chern, and Jean-Pierre Serre.
Derivation and proof
Proofs follow the analytic approaches of Atiyah, Singer, and Michael Atiyah's collaborators, or the representation-theoretic methods associated with Élie Cartan and Claude Chevalley. One computes local expressions using gamma matrices introduced by Paul Dirac and Hermann Weyl, applies Clifford algebra identities developed in work of Élie Cartan and Friedrich Hirzebruch, and compares connections in the style of Levi-Civita and Élie Cartan. Alternative proofs exploit heat kernel techniques popularized by Atiyah, Singer, and Daniel Quillen, or index-theoretic arguments reminiscent of Atiyah–Bott localization and Bott periodicity discovered by Raoul Bott and Michel Atiyah.
Applications in geometry and physics
The formula underpins vanishing theorems used by Lichnerowicz and later by Jean-Pierre Serre, Shing-Tung Yau, and Simon Donaldson to study harmonic spinors and metrics with positive scalar curvature linked to work by John Milnor and Michael Freedman. In mathematical physics it informs supersymmetry analyses by Edward Witten, string theory considerations by Andrew Strominger and Cumrun Vafa, and black hole studies by Stephen Hawking and Roger Penrose. It is fundamental in proofs of rigidity theorems due to Gromov and Lawson, in treatments of Seiberg–Witten invariants developed by Witten and Clifford Taubes, and in relations to the Atiyah–Singer index theorem formulated by Atiyah and Singer.
Generalizations and related results
Generalizations include twisted Dirac operators studied by Atiyah, Singer, and Michael Atiyah with vector bundles investigated by Jean-Louis Koszul and Henri Cartan, Weitzenböck identities used by Bochner and Salamon, and noncommutative geometry extensions pursued by Alain Connes and Matilde Marcolli. Extensions to Dirac-type operators on manifolds with boundary connect to work by Atiyah, Patodi, and Singer, and analytical refinements employ heat kernel methods of Ray and Singer, and zeta-regularization techniques used by Stephen Hawking and Daniel Quillen. Relations to the Schrödinger–Lichnerowicz inequality echo investigations by Paul Dirac and Marcel Berger in Riemannian geometry.
Examples and computations
On the round sphere studied by Élie Cartan and Bernhard Riemann, explicit eigenvalues computed in the tradition of Hermann Weyl and Emil Artin illustrate the formula via harmonic spinors analyzed by Marcel Berger and Alfred Gray. For Kähler manifolds considered by Kunihiko Kodaira and Shing-Tung Yau, the Dolbeault operator analogues link to work by Jean-Pierre Serre, Kunihiko Kodaira, and André Weil. Computations in low dimensions draw on tools popularized by Michael Freedman, Simon Donaldson, and Edward Witten for four-manifolds classified by Freedman, and examples in general relativity reflect curvature analyses by Albert Einstein, Roger Penrose, and Stephen Hawking.