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We solve a problem posed by Calabi more than 60 years ago, known as theSaint-Venant compatibility problem: Given a compact Riemannian manifold,generally with boundary, find a compatibility operator for Lie derivatives ofthe metric tensor. This problem is related to other compatibility problems inmathematical physics, and to their inherent gauge freedom. To this end, wedevelop a framework generalizing the theory of elliptic complexes for sequencesof linear differential operators $(A_{\bullet})$ between sections of vectorbundles. We call such a sequence an elliptic pre-complex if the operatorssatisfy overdetermined ellipticity conditions, and the order of $A_{k+1}A_k$does not exceed the order of $A_k$. We show that every elliptic pre-complex$(A_{\bullet})$ can be "corrected" into a complex $(\mathcal{A}_{\bullet})$ ofpseudodifferential operators, where $\mathcal{A}_k - A_k$ is a zero-ordercorrection within this class. The induced complex $(\mathcal{A}_{\bullet})$yields Hodge-like decompositions, which in turn lead to explicit integrabilityconditions for overdetermined boundary-value problems, with uniqueness andgauge freedom clauses. We apply the theory on elliptic pre-complexes ofexterior covariant derivatives of vector-valued forms and double formssatisfying generalized algebraic Bianchi identities, thus resolving a set ofcompatibility and gauge problems, among which one is the Saint-Venant problem.

Elliptic Pre-Complexes, Hodge-like Decompositions and Overdetermined Boundary-Value Problems