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We provide a presymplectic characterization of Liouville sectors introducedby Ganatra-Pardon-Shende in terms of the characteristic foliation of theboundary, which we call Liouville $\sigma$-sectors. We extend this definitionto the case with corners using the presymplectic geometry of null foliations ofthe coisotropic intersections of transversal coisotropic collection ofhypersurfaces which appear in the definition of Liouville sectors with corners.We identify its automorphism group which enables one to give a naturaldefinition of bundles of Liouville sectors. As a byproduct, we affirmativelyanswer to a question raised by Ganatra-Pardon-Shende, which asks about theoptimality of their definition of Liouville sectors.

Presymplectic geometry and Liouville sectors with corners and its intrinsic characterization