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 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 $p$.

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.