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[edit]

If is a smooth function, then

,

where is the gradient 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[edit]

References[edit]

  1. ^ 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, MR 2274812.