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	~~The '''Lichnerowicz formula''' (also known as the '''Lichnerowicz–Weitzenböck formula''')~~ is ~~a fundamental equation in the analysis of [[spinor]]s on [[pseudo~~-~~Riemannian manifold]]s~~. In dimension 4, it forms a piece of [[Seiberg–Witten gauge theory\|Seiberg–Witten theory]] and other aspects of [[gauge theory]]. It is named after noted mathematicians [[André Lichnerowicz]] who proved it in 1963, and [[Roland Weitzenböck]]. The formula gives a relationship between the ~~[[Dirac operator]]~~ and ~~the [[Laplace–Beltrami operator]] acting on spinors, in which the [[scalar curvature]] appears in a natural way. The result is significant because~~ it ~~provides an interface between results from the study of [[elliptic partial differential equation]]s, results concerning the scalar curvature, and results on spinors and spin structures~~.		Subsequent ɑ bаlancеd diet is essential - but many people have different demands. Utilize the pursuing ideas to find out the thing you need and make sure you are acquiring it.<br><br>

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	:<~~math~~>~~D^2\psi = \nabla^*\nabla\psi + \frac{1}{4}\operatorname{Sc}\psi~~<~~/math~~>
	~~where Sc denotes~~ the ~~[[scalar curvature]]~~ and <~~math~~>~~\nabla^*\nabla~~<~~/math~~> is ~~the [[Lichnerowicz_Laplacian#Connection_Laplacian\|connection Laplacian]]~~. ~~More generally~~, ~~given~~ a ~~[[complex spin structure]] on a pseudo-Riemannian manifold~~ ''M'', a ~~[[spinor bundle]] ''W''~~<~~sup~~>±<~~/sup~~> ~~with section~~ <~~math~~>~~\phi~~<~~/math~~>, and a ~~connection ''A'' on its~~ [[~~determinant line bundle~~]~~] ''L''~~, the ~~Lichnerowicz formula is~~

	:~~<math>D_{A}^{}D_{A}\phi~~=~~\nabla _A^{}\nabla_{A}\phi+\frac{1}{4}R\phi+\frac{1}{2}\langle F_{A}^{+}~~,~~\phi\rangle~~.<~~/math~~>

	~~Here,~~ <~~math~~>~~D_A~~<~~/math~~> ~~is the [[Dirac operator]]~~ <~~math~~>~~D_A:\Gamma (W^+)\~~to ~~\Gamma (W^-)~~,~~</math>~~ and <~~math~~>~~\nabla _A~~<~~/math~~> is ~~the covariant derivative associated~~ with ~~the connection A~~, ~~<math>\nabla _A: \Gamma (W^+)\~~to ~~\Gamma(W^+\otimes T_M^*)</math>~~. <~~math~~>R<~~/math~~> ~~is the usual scalar curvature (~~a ~~contraction~~ of ~~the~~ [~~[Ricci curvature\| Ricci tensor~~]~~]) and~~ <~~math~~>~~F_A^+~~<~~/math~~> is ~~the~~ [~~[self~~-~~dual]] part of~~ the ~~curvature~~ of A. ~~The asterisks denote the adjoint~~ of ~~the quantity~~ and ~~the brackets~~ <~~math~~>~~\langle , \rangle~~<~~/math~~> ~~denote the [[Clifford algebra\|Clifford action]]~~.

	~~==See also==~~
	* [[Weitzenböck formula]]

	~~==References==~~
	* {{Citation \| last1=Lawson \| first1=H. Blaine \| last2=Michelsohn \| first2=Marie-Louise \|author2-link=Marie-Louise Michelsohn\| title=Spin Geometry \| publisher=[[Princeton University Press]] \| isbn=978-0-691-08542-5 \| year=1989}}
	* {{Citation \|last1=LeBrun\|first1=Claude\|authorlink= Claude LeBrun \|title=Einstein Metrics, 4-~~Manifolds & Differential Topology\|year=2002}}~~
	* {{Citation \| last1=Scorpan \| first1=Alexandru \| title=The Wild World of ~~4-Manifolds \| publisher=[[American Mathematical Society]] \| location=Providence, Rhode Island \| year=2005}}~~

	~~[[Category:Differential geometry]]~~
	~~[[Category:Riemannian geometry]]~~

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