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{{Merge|Bochner identity|date=October 2007}} |
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{{Merge|Bochner identity|date=October 2007}} |
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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> |
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In ], '''Bochner's formula''' : <math> |
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\triangle f = |\nabla X|^2 - Ric(X,X) |
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\triangle \frac{1}{2}|\nabla u| ^2 = |\nabla^2 u|^2 - \mbox{Ric}(\nabla u, \nabla u) |
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</math>. The formula is an example of a ]. Bochner used this formula to prove the ]. |
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<\math>, where <math> f = \frac{1}{2}g(X,X) <\math>, |
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is a significant result of ] in ]. Loosely speaking, it says that the difference between the two ]-like operators on the tangent bundle of a ] is a zero-order operator determined by the Ricci curvature. It is an example of a ]. Bochner used this formula to prove the ]. |
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The Bochner formula is often proved using ] or ] methods. |
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The Bochner formula is often proved using ] or ] methods. |