Bochner's formula
In mathematics, Bochner's formula is a statement relating harmonic functions on a Riemannian manifold [math]\displaystyle{ (M, g) }[/math] to the Ricci curvature. The formula is named after the United States mathematician Salomon Bochner.
Formal statement
If [math]\displaystyle{ u \colon M \rightarrow \mathbb{R} }[/math] is a smooth function, then
- [math]\displaystyle{ \tfrac12 \Delta|\nabla u|^2 = g(\nabla\Delta u,\nabla u) + |\nabla^2 u|^2 + \mbox{Ric}(\nabla u, \nabla u) }[/math],
where [math]\displaystyle{ \nabla u }[/math] is the gradient of [math]\displaystyle{ u }[/math] with respect to [math]\displaystyle{ g }[/math], [math]\displaystyle{ \nabla^2 u }[/math] is the Hessian of [math]\displaystyle{ u }[/math] with respect to [math]\displaystyle{ g }[/math] and [math]\displaystyle{ \mbox{Ric} }[/math] is the Ricci curvature tensor.[1] If [math]\displaystyle{ u }[/math] is harmonic (i.e., [math]\displaystyle{ \Delta u = 0 }[/math], where [math]\displaystyle{ \Delta=\Delta_g }[/math] is the Laplacian with respect to the metric [math]\displaystyle{ g }[/math]), Bochner's formula becomes
- [math]\displaystyle{ \tfrac12 \Delta|\nabla u| ^2 = |\nabla^2 u|^2 + \mbox{Ric}(\nabla u, \nabla u) }[/math].
Bochner used this formula to prove the Bochner vanishing theorem.
As a corollary, if [math]\displaystyle{ (M, g) }[/math] is a Riemannian manifold without boundary and [math]\displaystyle{ u \colon M \rightarrow \mathbb{R} }[/math] is a smooth, compactly supported function, then
- [math]\displaystyle{ \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} }[/math].
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
- ↑ Chow, Bennett; Lu, Peng; Ni, Lei (2006), Hamilton's Ricci flow, Graduate Studies in Mathematics, 77, Providence, RI: Science Press, New York, p. 19, ISBN 978-0-8218-4231-7, https://books.google.com/books?id=T1K5fHoRalYC&pg=PA19.
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