Bochner's formula

Updated on Nov 26, 2023

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In mathematics, Bochner's formula is a statement relating harmonic functions on a Riemannian manifold ( M , g ) to the Ricci curvature.

Formal statement

More specifically, if u : M → R is a harmonic function (i.e., Δ g u = 0 , where Δ g is the Laplacian with respect to g ), then

Δ 1 2 | ∇ u | 2 = | ∇ 2 u | 2 + Ric ( ∇ u , ∇ u ) ,

where ∇ u is the gradient of u with respect to g . Bochner used this formula to prove the Bochner vanishing theorem.

Bochner identity

Weitzenböck identity.

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