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{{technical|date=June 2012}}
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In ], '''Bochner's formula''' is a statement relating ] on a ] <math> (M, g) </math> to the ]. More specifically, if <math> u : (M, g) \rightarrow \mathbb{R} </math> is a harmonic function, so <math> \triangle_g u = 0 </math> (<math> \triangle </math> is the Laplacian operator), then <math> In ], '''Bochner's formula''' is a statement relating ] on a ] <math> (M, g) </math> to the ]. The formula is named after the ] ] ].
⚫ \triangle \frac{1}{2}|\nabla u| ^2 = |\nabla^2 u|^2 - \mbox{Ric}(\nabla u, \nabla u)
⚫ </math>. The formula is an example of a ]. Bochner used this formula to prove the ].


==Formal statement==
The Bochner formula is often proved using ] or ] methods.
If <math> u \colon M \rightarrow \mathbb{R} </math> is a smooth function, then
:<math>
\tfrac12 \Delta|\nabla u|^2 = g(\nabla\Delta u,\nabla u) + |\nabla^2 u|^2 + \mbox{Ric}(\nabla u, \nabla u)
</math>,
where <math> \nabla u </math> is the ] of <math>u</math> with respect to <math> g</math>, <math> \nabla^2 u </math> is the ] of <math>u</math> with respect to <math> g</math> and <math> \mbox{Ric} </math> is the ].<ref>{{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 = ]
| title = Hamilton's Ricci flow
| url = https://books.google.com/books?id=T1K5fHoRalYC&pg=PA19
| volume = 77
| year = 2006}}.</ref> If <math> u </math> is harmonic (i.e., <math> \Delta u = 0 </math>, where <math> \Delta=\Delta_g </math> is the ] with respect to the metric <math> g </math>), Bochner's formula becomes
:<math>
⚫ \tfrac12 \Delta|\nabla u| ^2 = |\nabla^2 u|^2 + \mbox{Ric}(\nabla u, \nabla u)
</math>.
⚫ Bochner used this formula to prove the ].


As a corollary, if <math> (M, g) </math> is a Riemannian manifold without boundary and <math> u \colon M \rightarrow \mathbb{R} </math> is a smooth, compactly supported function, then
⚫ ]
:<math>
\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 ]) and integrating by parts the first term on the right-hand side.


==Variations and generalizations==
{{geometry-stub}}
*]
*]

==References==
{{reflist}}

⚫ ]

Latest revision as of 21:41, 7 September 2021

This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2012) (Learn how and when to remove this message)

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

Formal statement

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

1 2 Δ | ∇ u | 2 = g ( ∇ Δ u , ∇ u ) + | ∇ 2 u | 2 + Ric ( ∇ u , ∇ u ) {\displaystyle {\tfrac {1}{2}}\Delta |\nabla u|^{2}=g(\nabla \Delta u,\nabla u)+|\nabla ^{2}u|^{2}+{\mbox{Ric}}(\nabla u,\nabla u)} ,

where ∇ u {\displaystyle \nabla u} is the gradient of u {\displaystyle u} with respect to g {\displaystyle g} , ∇ 2 u {\displaystyle \nabla ^{2}u} is the Hessian of u {\displaystyle u} with respect to g {\displaystyle g} and Ric {\displaystyle {\mbox{Ric}}} is the Ricci curvature tensor. If u {\displaystyle u} is harmonic (i.e., Δ u = 0 {\displaystyle \Delta u=0} , where Δ = Δ g {\displaystyle \Delta =\Delta _{g}} is the Laplacian with respect to the metric g {\displaystyle g} ), Bochner's formula becomes

1 2 Δ | ∇ u | 2 = | ∇ 2 u | 2 + Ric ( ∇ u , ∇ u ) {\displaystyle {\tfrac {1}{2}}\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 ) {\displaystyle (M,g)} is a Riemannian manifold without boundary and u : M → R {\displaystyle u\colon M\rightarrow \mathbb {R} } is a smooth, compactly supported function, then

∫ M ( Δ u ) 2 d vol = ∫ M ( | ∇ 2 u | 2 + Ric ( ∇ u , ∇ u ) ) d vol {\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}}} .

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

  1. Chow, Bennett; Lu, Peng; Ni, Lei (2006), Hamilton's Ricci flow, Graduate Studies in Mathematics, vol. 77, Providence, RI: Science Press, New York, p. 19, ISBN 978-0-8218-4231-7, MR 2274812.
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