Notes

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====== Laplacian ======


Let $(M,g)$ be a Riemannian manifold, and $E \to M$ a vector bundle over $M$ with fibre metric $h$ and compatible connection $\nabla$.
The Laplace operator $\Delta : \Omega^k(E) \to \Omega^k(E)$ is defined on $E$-valued forms by
\[ \Delta = \mathrm{d}\mathrm{d}^* + \mathrm{d}^* \mathrm{d}. \]

===== Decomposing the Laplacian =====

Let $e_i$ be an orthonormal frame of $TM$ about some point $p$, with coframe $\omega^i$. We compute as follows
\begin{align*}
\Delta &= \mathrm{d}\mathrm{d}^* + \mathrm{d}^* \mathrm{d} \\
&= \left[-e(\omega^i)\nabla_i e^*(\omega^j)\nabla_j\right] - \left[e^*(\omega^j)\nabla_j e(\omega^i)\nabla_i\right] \\
&= \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] \\
&= \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] \\
&= \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*}

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
\[ \Delta \omega = \nabla^* \nabla \omega + \mathrm{Ric}(\omega, e_i)\omega^i. \]