Topology of singular foliations of closed 1-forms on orbifolds

TL;DR

This paper investigates the topological structure of leaves in singular foliations induced by Morse-type closed 1-forms on compact orbifolds, providing criteria for leaf compactness and extending classical results to orbifolds.

Contribution

It introduces new topological criteria for leaf properties in orbifold foliations and generalizes Calabi's characterization of closed harmonic 1-forms to orbifolds.

Findings

Criteria for leaves being compact or non-compact

Existence of coexistence of different leaf types

Extension of Calabi's theorem to orbifold setting

Abstract

We study the topological properties of the leaves of the singular foliation induced by a closed 1-form of Morse type on a compact orbifold. In particular, we establish criteria that characterize when all such leaves are compact, when they are non-compact, and how both types may coexist. As an application, we extend to the orbifold setting a celebrated result of Calabi, which provides a purely topological characterization of intrinsically closed harmonic 1-forms of Morse type.