Bochner's formula
In mathematics, Bochner's formula is a statement relating harmonic functions on a Riemannian manifold formula_1 to the Ricci curvature. The formula is named after the American mathematician Salomon Bochner. Formal statement. If formula_2 is a smooth function, then where formula_4 is the gradient of formula_5 with respect to formula_6, formula_7 is the Hessian of formula_5 with respect to formula_6 and formula_10 is the Ricci curvature tensor. If formula_11 is harmonic (i.e., formula_12, where formula_13 is the Laplacian with respect to the metric formula_14), Bochner's formula becomes Bochner used this formula to prove the Bochner vanishing  theorem. As a corollary, if formula_1 is a Riemannian manifold without boundary and formula_2 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.