an:06997262
Zbl 1406.58015
Bergamasco, Adalberto Panobianco; Parmeggiani, Alberto; Zani, S??rgio Lu??s; Zugliani, Giuliano Angelo
Geometrical proofs for the global solvability of systems
EN
Math. Nachr. 291, No. 16, 2367-2380 (2018).
00424037
2018
j
58J10 35A01 35N10
complex vector fields; global solvability; involutive systems
Summary: We study a linear operator associated with a closed non-exact 1-form \(b\) defined on a smooth closed orientable surface \(M\) of genus \(g>1\). Here we present two proofs that reveal the interplay between the global solvability of the operator and the global topology of the surface. The first result brings an answer for the global solvability when the system is defined by a generic Morse 1-form. Necessary conditions for the global solvability bearing on the sublevel and superlevel sets of primitives of a smooth 1-form \(b\) have already been established; we also present a more intuitive proof of this result.