In mathematics, Bochner's formula is a statement relating harmonic functions on a Riemannian manifold to the Ricci curvature. The formula is named after the American mathematician Salomon Bochner.
Formal statement If is a smooth function, then , where is the gradient of with respect to , is the Hessian of with respect to and is the Ricci curvature tensor.. If is harmonic (i.e., , where is the Laplacian with respect to the metric ), Bochner's formula becomes . Bochner used this formula to prove the Bochner vanishing theorem.
As a corollary, if 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 Bochner identity Weitzenböck identity
References