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--- laplacian [2022/09/08 12:40] – spencer
+++ laplacian [2022/09/10 15:32] (current) – [Decomposing the Laplacian] spencer
@@ Line 6: / Line 6: @@
 \[ \Delta = \mathrm{d}\mathrm{d}^* + \mathrm{d}^* \mathrm{d}. \]
+===== Decomposing the Laplacian =====
-==== Rough Laplacian ====
+Let $e_i$ be an orthonormal frame of $TM$ about some point $p$, with coframe $\omega^i$. We compute as follows
-Another Laplacian on forms is the rough Laplacian, $\nabla^* \nabla$.
-In an orthonormal frame $(e_i)_{i=1}^n$ of $TM$ with dual frame $(\omega^i)_{i=1}^n$, the rough Laplacian may be expressed in coordinates by
 \begin{align*}
-\nabla^* \nabla &= \mathrm{d}^* \mathrm{d} \\
+\Delta &= \mathrm{d}\mathrm{d}^* + \mathrm{d}^* \mathrm{d} \\
-&= -e^*(\omega^i)\nabla_{e_i} e(\omega^j) \nabla_{e_j} \\
+&= \left[-e(\omega^i)\nabla_i e^*(\omega^j)\nabla_j\right] - \left[e^*(\omega^j)\nabla_j e(\omega^i)\nabla_i\right] \\
-&= -e^*(\omega^i) e(\omega^j) \nabla_{e_i} \nabla_{e_j} + e(\nabla_{e_i} \omega^j) \nabla_{e_j} \\
+&= \left[-e(\omega^i)(e^*(\nabla_i \omega^j) + e^*(\omega^j)\nabla_i)\nabla_j\right] - \left[e^*(\omega^j)(e(\nabla_j \omega^i) + e(\omega^i)\nabla_j)\nabla_i \right] \\
-&= -\nabla_{e_i} \nabla_{e_i} + (-e^*(\omega^i) e(\nabla_{e_i} e_j) \nabla_{e_j}) \\
+&= \left[-e(\omega^i)e^*(\nabla_i \omega^j)\nabla_j - e(\omega^i)e^*(\omega^j)\nabla_i\nabla_j\right] - \left[e^*(\omega^j)e(\nabla_j \omega^i)\nabla_i + e^*(\omega^j)e(\omega^i)\nabla_j\nabla_i \right] \\
-&= -\nabla_{e_i} \nabla_{e_i} - \langle \nabla_{e_i} e_j, e_i \rangle \nabla_{e_j}.
+&= \left[-e(\omega^i)e^*(\nabla_i \omega^j)\nabla_j - e(\omega^i)e^*(\omega^j)\nabla_i\nabla_j\right] - \left[\langle \nabla_j \omega^i, \omega^j \rangle \nabla_i - e(\nabla_j \omega^i) e^*(\omega^j) \nabla_i + \langle \omega^j, \omega^i\rangle\nabla_j\nabla_i - e(\omega^i)e^*(\omega^j) \nabla_j \nabla_i \right] \\
+&= \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) + 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) - e(\omega^i) e^*(\omega^j) \nabla_{[e^i, e^j]} \\
+&= \nabla^* \nabla + e(\omega^i) e^*(\omega^j) R(e_i, e_j).
 \end{align*}
-To simplify further, remark that
-\begin{align*}
-\langle \nabla_{e_i} e_j, e_i \rangle_{L^2} \nabla_{e_j} &= -\langle e_j, \nabla_{e_i} e_i \rangle_{L^2} \nabla_{e_j} \\
-&= - \nabla_{\langle e_j, \nabla_{e_i} e_i\rangle_{L^2} e_j} \\
-&= -\nabla_{\nabla_{e_i} e_i},
-\end{align*}
-and so
-\begin{align*}
-\nabla^* \nabla &= -\nabla_{e_i} \nabla_{e_i} + \nabla_{\nabla_{e_i} e_i}.
-\end{align*}
-===== Difference between Laplacians =====
-By the above computation of the rough Laplacian,
-\begin{align*}
-\Delta &= \nabla^* \nabla + \mathrm{d}\mathrm{d}^* \\
-&= \nabla^* \nabla - e(\omega^i) \nabla_{e_i} e^*(\omega^j) \nabla_{e_j}.
-\end{align*}
-Then
-\[ e(\omega^i) \nabla_{e_i} = \nabla_{e_i} e(\omega^i) - e(\nabla_{e_i} \omega^i), \]
-so
-\[ e(\omega^i)\nabla_{e_i} e^*(e_j) = \nabla_{e_j} - \nabla_{e_i} e(\nabla_{e_i} \omega^i) e^*(e_j). \]
-Recall that for any 1-form $\omega$ and vector field $X$ we have
-\[ e^*(X) e(\omega) = e^*(X)(\omega) - e(\omega)e^*(X). \]
-Rearranging, thus
-\begin{align*}
-e(\nabla_{e_i} \omega^i) e^*(e_j) &= (\nabla_{e_i} \omega^i)(e_j) - e^*(e_j)e(\nabla_{e_i} \omega^i) \\
-&= \omega^i(\nabla_{e_i} e_j) - \langle \nabla_{e_i} \omega^i, e_j \rangle \\
-&= \langle \nabla_{e_i} e_j, e_i \rangle - \langle \nabla_{e_i} e_i, e_j \rangle \\
-\end{align*}
-Something looks wrong here.
+The action of the curvature on a 1-form $\omega$ is given by
+\[ (R(X,Y)\omega)(Z) = R(X,Y) (\omega(Z)) - \omega(R(X,Y)Z) = -\omega(R(X,Y)Z). \]
+So
+\[ e(\omega^i) e^*(\omega^j) R(e_i,e_j) \omega = -\omega(R(e_i, e_j) e_j) \omega^i = \mathrm{Ric}(\omega, e_i)\omega^i,\]
+whence
+\[ \Delta \omega = \nabla^* \nabla \omega + \mathrm{Ric}(\omega, e_i)\omega^i. \]