Bochner's formula

Remarks

The idea of the proof: compute at a point in normal coordinates. The left side is $\sum_{i,j} (\nabla_i \nabla_j u)^2 + \sum_{i,j} \nabla_j u \, \nabla_i \nabla_i \nabla_j u$. Commuting the covariant derivatives in the last term gives $\nabla_j \nabla_i \nabla_i u$ plus a curvature term, and that curvature term is exactly $\operatorname{Ric}(\nabla u, \nabla u)$.

Example: in $\R^n$ the Ricci term is $0$. For $u(x) = \abs{x}^2 / 2$ one has $\nabla u = x$, $\nabla^2 u = \id$ and $\Delta u = n$, so the left side is $\frac{1}{2} \Delta \abs{x}^2 = n$ and the right side is $n + 0 + 0$.

A standard application: let $M$ be compact of dimension $n$ with $\operatorname{Ric} \ge (n-1) k \, g$ for some $k > 0$. Integrating the formula for an eigenfunction shows that every nonzero eigenvalue $\lambda$ of $-\Delta$ satisfies $\lambda \ge n k$. This is Lichnerowicz's eigenvalue estimate.

Going deeper

For a student

Bochner's formula is the main identity behind the "Bochner technique": a curvature sign forces certain objects to be parallel or to vanish. It is used in the Cheng–Yau gradient estimate, in the Laplacian comparison theorem, in the proof of the Cheeger–Gromoll splitting theorem and in eigenvalue estimates. The same computation for 1-forms is the case of the Weitzenböck identity that leads to Bochner's vanishing theorem for the first Betti number.