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On global hypoellipticity. (English) Zbl 1387.58037
Summary: We consider a first order linear partial differential operator of principal type on a closed connected orientable two-dimensional manifold sending sections of one complex line bundle to sections of another. We prove that the assumption of global hypoellipticity of the operator implies a relation between the degrees of the line bundles and the Euler characteristic of the manifold.

MSC:
58J99 Partial differential equations on manifolds; differential operators
35H10 Hypoelliptic equations
35B65 Smoothness and regularity of solutions to PDEs
35F05 Linear first-order PDEs
57R22 Topology of vector bundles and fiber bundles
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