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Global solvability of real analytic involutive systems on compact manifolds. (English) Zbl 1380.35129
Let \(b\) be a real analytic closed non-exact 1-form defined on a compact and without boundary connected \(n\)-dimensional manifold \(M\), \((n > 1)\). The focus of this work is the smooth global solvability of the differential operator \({\mathbb L}:C^\infty(M\times {\mathbb S}^1)\to \Lambda^1 C^\infty(M\times {\mathbb S}^1)\) given by \({\mathbb L}u =d_{t_j}u + ib(t) \wedge \partial_x u\), where \(x\in {\mathbb S}^1\), and \(d_t\) is the exterior derivative on \(M\).
The approach relies on defining an appropriate covering projection \(\tilde M \to M\) such that the pullback of \(b\) has a primitive \(\tilde B\) and prove that the operator is globally solvable if and only if the superlevel and sublevel sets of \(\tilde B\) are connected. In case of orientable manifolds \(M\), further charactirizations are made.

MSC:
35N10 Overdetermined systems of PDEs with variable coefficients
58J10 Differential complexes
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