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--- weitzenboeck_identity [2022/09/03 06:19] – created 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:
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 \[ S_p(\sigma(X_1,\ldots,X_k)) = \sum_{s,k} (-1)^k (\iota_{e_s} (R(e_s, X_k)\sigma))(X_1,\ldots, \widehat{X_k},\ldots, X_p), \]
 with the hat denoting omission and $\iota$ the interior product.
+One consequence of the Weitzenböck formula is a nice formula for the Laplacian of the norm of a $k$-form:
+\[ \frac{1}{2} \Delta |\sigma|^2 = \langle \Delta \sigma, \sigma\rangle - |\nabla \sigma|^2 - \langle S(\sigma), \sigma \rangle. \]
+In particular, if $\sigma$ is a 1-form, then
+\begin{align*}
+ S_p(\sigma(X)) &= -\sum_s (R(e_s,X)\sigma)(e_s) \\
+&= \sum_s \sigma(R^M(e_s, X)e_s) - R^E(e_s, X)(\sigma(e_s)),
+\end{align*}
+and so
+\[ \frac{1}{2} \Delta |\sigma|^2 = \langle \Delta \sigma, \sigma \rangle - |\nabla \sigma|^2 + \sum_{s,t} \langle R^E(e_s, e_t)\sigma(e_s), \sigma(e_t)\rangle - \sum_t \langle \sigma(\mathrm{Ric}(e_t)), \sigma(e_t)\rangle. \]
+Finally, if $\sigma$ is a //harmonic// 1-form, then the identity reads
+\[ \frac{1}{2} \Delta |\sigma|^2 =- |\nabla \sigma|^2 + \sum_{s,t} \langle R^E(e_s, e_t)\sigma(e_s), \sigma(e_t)\rangle - \sum_t \langle \sigma(\mathrm{Ric}(e_t)), \sigma(e_t)\rangle. \]
+Setting $\sigma = \mathrm{d}u$ for a harmonic map $u$ gives precisely the **Bochner identity**.