- Formal statement
- Variations and generalizations
- References
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
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
- Bochner identity
- Weitzenböck identity
References
- Van der Zee
- Vanderzwaan
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- Van de Sompel
- Van de Spiegheling der singconst
- Van Dessel
- Van Dessel Sports
- VanDeSteeg
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- Vandeurzen (disambiguation)
- Van Deurzen (disambiguation)
- Van Deusen's Rat
- Vandev
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- Van Deven
- Van de Ven
- Van de Ven, A.
- Van de Ven, Andrew H.
- Van de Venne