Bochner's formula explained

(M, g) to the Ricci curvature. The formula is named after the American mathematician Salomon Bochner.

Formal statement

If

u \colon M → R is a smooth function, then\tfrac 12 \Delta|\nabla u| ^{2} = g(\nabla\Delta u,\nabla u) + |\nabla ^{2} u| ^{2} + Ric (\nabla u, \nabla u) ,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 Ric is the Ricci curvature tensor.^[1] If u is harmonic (i.e., \Delta u = 0 , where \Delta = \Delta _{g} is the Laplacian with respect to the metric g ), Bochner's formula becomes\tfrac 12 \Delta|\nabla u| ^{2} = |\nabla ^{2} u| ^{2} + Ric (\nabla u, \nabla u) .Bochner used this formula to prove the Bochner vanishing theorem.

As a corollary, if

(M, g) is a Riemannian manifold without boundary and u \colon M → R is a smooth, compactly supported function, then\int _{M} (\Delta u) ^{2} d vol = \int _{M} ( |\nabla ^{2} u| ^{2} + Ric (\nabla u, \nabla u) ) d vol .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

Notes and References

1. .