In mathematics, Bochner's formula is a statement relating harmonic functions on a Riemannian manifold (M, g) to the Ricci curvature. The formula is named after the American mathematician Salomon Bochner.

Formal statement

If is a smooth function, then where \nabla u is the gradient of u with respect to g, \nabla^2 u is the Hessian of u with respect to g and \mbox{Ric} is the Ricci curvature tensor. If u is harmonic (i.e.,, where is the Laplacian with respect to the metric g), Bochner's formula becomes Bochner used this formula to prove the Bochner vanishing theorem. As a corollary, if (M, g) is a Riemannian manifold without boundary and is a smooth, compactly supported function, then This immediately follows from the first identity, observing that the integral of the left-hand side vanishes (by the divergence theorem) and integrating by parts the first term on the right-hand side.

Variations and generalizations