Bochner's Formula

Bochner's formula is a result in differential geometry, named after the mathematician Salomon Bochner, that relates harmonic functions on a Riemannian manifold to the manifold's Ricci curvature. For a smooth function u on a Riemannian manifold, the formula states that one half of the Laplacian of the squared length of the gradient of u equals the inner product of the gradient of the Laplacian of u with the gradient of u, plus the squared norm of the Hessian of u, plus the Ricci curvature evaluated on the gradient of u. When u is harmonic, meaning its Laplacian is zero, this relation simplifies and directly ties the geometry of the gradient to the curvature of the underlying manifold. Bochner used this formula to prove what is now called the Bochner vanishing theorem, and integrating the formula by parts yields further results connecting curvature to the global geometry of the manifold. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/