Bochner's formula

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

Formal statement

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If   is a smooth function, then

 ,

where   is the gradient of   with respect to  ,   is the Hessian of   with respect to   and   is the Ricci curvature tensor.[1] If   is harmonic (i.e.,  , where   is the Laplacian with respect to the metric  ), Bochner's formula becomes

 .

Bochner used this formula to prove the Bochner vanishing theorem.

As a corollary, if   is a Riemannian manifold without boundary and   is a smooth, compactly supported function, then

 .

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

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References

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  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.