Differences

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--- laplacian [2022/09/09 09:19] – spencer
+++ laplacian [2022/09/10 15:32] (current) – [Decomposing the Laplacian] spencer
@@ Line 17: / Line 17: @@
 &= \left[-e(\omega^i)e^*(\nabla_i \omega^j)\nabla_j - e(\omega^i)e^*(\omega^j)\nabla_i\nabla_j\right] - \left[-\nabla_{\nabla_i e_i} - e(\nabla_j \omega^i) e^*(\omega^j) \nabla_i + \nabla_i \nabla_i - e(\omega^i)e^*(\omega^j) \nabla_j \nabla_i \right] \\
 &= -\nabla_i \nabla_i + \nabla_{\nabla_i e_i} + e(\omega^i)e^*(\omega^j)(\nabla_j \nabla_i - \nabla_i \nabla_j) + e(\nabla_j \omega^i)e^*(\omega^j)\nabla_i - e(\omega^i)e^*(\nabla_i \omega^j) \nabla_j \\
-&= \nabla^* \nabla + e(\omega^i)e^*(\omega^j)(\nabla_j \nabla_i - \nabla_i \nabla_j) + \left[e(\nabla_j \omega^i)e^*(\omega^j) - e(\omega^j)e^*(\nabla_j \omega^i)\right] \nabla_i \\
+&= \nabla^* \nabla + e(\omega^i)e^*(\omega^j) (\nabla_j \nabla_i - \nabla_i \nabla_j) + e(\omega^i) e^*(\omega^j) (\nabla_{\nabla_j e_i} - \nabla_{\nabla_i e_j}) \\
-&= \nabla^* \nabla + e(\omega^i)e^*(\omega^j)(\nabla_j \nabla_i - \nabla_i \nabla_j) + \left[e^*(\omega^j)e(\nabla_j \omega^i) - e^*(\nabla_j \omega^i)e(\omega^j)\right] \nabla_i \\
+&= \nabla^* \nabla + e(\omega^i)e^*(\omega^j) (\nabla_j \nabla_i - \nabla_i \nabla_j) - e(\omega^i) e^*(\omega^j) \nabla_{[e^i, e^j]} \\
-&= \nabla^* \nabla + e(\omega^i)e^*(\omega^j)(\nabla_j \nabla_i - \nabla_i \nabla_j).
+&= \nabla^* \nabla + e(\omega^i) e^*(\omega^j) R(e_i, e_j).
 \end{align*}
-Acting on a 1-form $\omega$, then
+The action of the curvature on a 1-form $\omega$ is given by
-\[ e(\omega^i)e^*(\omega^j)(\nabla_j \nabla_i - \nabla_i \nabla_j)\omega = e(\omega^i)e^* (\omega^j)R^{TM \otimes E}_{i,j} \omega = e(\omega^i)e^*(\omega^j)(R^{TM} + R^E)\omega.\]
+\[ (R(X,Y)\omega)(Z) = R(X,Y) (\omega(Z)) - \omega(R(X,Y)Z) = -\omega(R(X,Y)Z). \]
-In the $M$-part,
+So
-\[ e(\omega^i)e^*(\omega^j) R^{TM} \omega = \]
+\[ e(\omega^i) e^*(\omega^j) R(e_i,e_j) \omega = -\omega(R(e_i, e_j) e_j) \omega^i = \mathrm{Ric}(\omega, e_i)\omega^i,\]
+whence
+\[ \Delta \omega = \nabla^* \nabla \omega + \mathrm{Ric}(\omega, e_i)\omega^i. \]