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====== Weitzenböck identity ======

Let $E \to M$ be a vector bundle with connection $\nabla$ over a compact Riemannian manifold.
There are (up to a sign) two natural Laplacians one may place on the space $\Omega^{\bullet}(E)$ of $E$-valued differential forms on $M$.
The first is the Hodge Laplacian $\Delta = \mathrm{d}\mathrm{d}^* + \mathrm{d}^* \mathrm{d}$, where $\mathrm{d}$ is the [[exterior_derivative|exterior derivative]] induced by $\nabla$ and $\mathrm{d}^*$ is its adjoint with respect to the $L^2$-metric on the space of $E$-valued forms.

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 $p$; 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:
\[ \Delta = \nabla^* \nabla + S, \]
where for each $k$, $S$ is an endomorphism of $\Lambda^k T^*M \otimes E$.

The curvature $R$ of $\Lambda^k T^*M \otimes E$ is given by
\[(R(X,Y)\sigma) = R^E(X,Y)(\sigma(X_1,\ldots,X_k)) - \sum_{i=1}^k \sigma(X_1,\ldots, R^M(X,Y)X_i, \ldots, X_k), \]
where $R^E, R^M$ respectively denote the curvature tensors on the bundles $E, TM$, and $X, Y, X_i$ are vector fields on $M$.
Then at a point $p \in M$ with a local orthonormal frame $(e_i)_{i=1}^n$, one has precisely that
\[ S_p(\sigma(X_1,\ldots,X_k)) = \sum_{s,k} (-1)^k (\iota_{e_s} (R(e_s, X_k)\sigma))(X_1,\ldots, \widehat{X_k},\ldots, X_p), \]
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**.