Lichnerowicz-Weitzenbock Formula

$$D^*D\psi=\nabla^*\nabla\psi+{\textstyle{1\over 4}}R\psi,$$

where $D$ is the Dirac operator $D:\Gamma(S^+)\to\Gamma(S^-)$, $\nabla$ is the Covariant Derivative on Spinors, and $R$ is the Curvature Scalar.

See also Lichnerowicz Formula

References

Donaldson, S. K. ``The Seiberg-Witten Equations and 4-Manifold Topology.'' Bull. Amer. Math. Soc. 33, 45-70, 1996.