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.
is a smooth function, then where
with respect to
with respect to
is the Ricci curvature tensor.
is the Laplacian with respect to the metric
), Bochner's formula becomes Bochner used this formula to prove the Bochner vanishing theorem.
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.