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 $u \colon M \rightarrow \mathbb$ is a smooth function, then
:$\tfrac12 \Delta, \nabla u, ^2 = g(\nabla\Delta u,\nabla u) + , \nabla^2 u, ^2 + \mbox(\nabla u, \nabla u)$,
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 $\mbox$ is the Ricci curvature tensor.. 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
:$\tfrac12 \Delta, \nabla u, ^2 = , \nabla^2 u, ^2 + \mbox(\nabla u, \nabla u)$.
Bochner used this formula to prove the Bochner vanishing theorem.

As a corollary, if $(M, g)$ is a Riemannian manifold without boundary and $u \colon M \rightarrow \mathbb$ is a smooth, compactly supported function, then
:$\int_M (\Delta u)^2 \, d\mbox = \int_M \Big( , \nabla^2 u, ^2 + \mbox(\nabla u, \nabla u) \Big) \, d\mbox$.
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

References

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Differential geometry