Differences

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--- weitzenboeck_identity [2022/09/03 06:36] – spencer
+++ weitzenboeck_identity [2022/09/05 05:36] (current) – spencer
@@ Line 6: / Line 6: @@
 The second is the Bochner Laplacian $\nabla^* \nabla$, where $\nabla : \Omega^k(E) \to \Omega^1(M) \otimes \Omega^k(E)$ (see [[section_isomorphisms]]) is the induced connection on $\Lambda^k T^*M \otimes E$.
-This differs by a sign from the Laplace–Beltrami operator $\mathrm{tr}\, \nabla^2$ which traces with respect to the metric on $M$ the bilinear form $T_p M \times T_p M \to (\Lambda^k T^*M \otimes E)_p$ given pointwise by two applications of the connection to an $E$-valued $k$-form at a point $p$.
+This differs by a sign from the Laplace–Beltrami operator $\mathrm{tr}\, \nabla^2$ which traces with respect to the metric on $M$ the bilinear form $T_p M \times T_p M \to (\Lambda^k T^*M \otimes E)_p$ given pointwise by two applications of the connection to an $E$-valued $k$-form at a point $p$; indeed, in a local orthonormal frame,
+\[ \mathrm{tr} \nabla^2 \omega = \sum (\nabla^2 \omega)(e_i, e_i) = e(\omega^i)^* \nabla_{e_i} \omega \]
+is precisely the negative of the adjoint to the [[exterior_derivative|exterior derivative]].
 The Weitzenböck identity relates these two Laplacians as follows: