Bochner's formula

In mathematics, Bochner's formula is a statement relating harmonic functions on a Riemannian manifold $(M, g)$ to the Ricci curvature. The formula is named after the American mathematician Salomon Bochner.

Formal statement

If $u \colon M \rightarrow \mathbb{R}$ is a smooth function, then

: $\tfrac12 \Delta|\nabla u|^2 = g(\nabla\Delta u,\nabla u) + |\nabla^2 u|^2 + \mbox{Ric}(\nabla u, \nabla u)$ ,

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 $\mbox{Ric}$ is the Ricci curvature tensor.{{citation

| last1 = Chow | first1 = Bennett

| last2 = Lu | first2 = Peng

| last3 = Ni | first3 = Lei

| isbn = 978-0-8218-4231-7

| location = Providence, RI

| mr = 2274812

| page = 19

| publisher = Science Press, New York

| series = Graduate Studies in Mathematics

| title = Hamilton's Ricci flow

| url = https://books.google.com/books?id=T1K5fHoRalYC&pg=PA19

| volume = 77

| year = 2006}}. 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

: $\tfrac12 \Delta|\nabla u| ^2 = |\nabla^2 u|^2 + \mbox{Ric}(\nabla u, \nabla u)$ .

Bochner used this formula to prove the Bochner vanishing theorem.

As a corollary, if $(M, g)$ is a Riemannian manifold without boundary and $u \colon M \rightarrow \mathbb{R}$ is a smooth, compactly supported function, then

: $\int_M (\Delta u)^2 \, d\mbox{vol} = \int_M \Big( |\nabla^2 u|^2 + \mbox{Ric}(\nabla u, \nabla u) \Big) \, d\mbox{vol}$ .

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

References

Category:Differential geometry