Topology of singular foliations of closed 1-forms on orbifolds

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.

Figures (6)

• Figure 1: Singular foliation in $\mathbb{R}^3$ defined by $x^2+y^2-z^2=t$ at $t=-2, 0, 2$, respectively.
• Figure 2: Singular foliation in $\mathbb{R}^3/\mathbb{Z}_2$ defined by $x^2+y^2-z^2=t$ at $t=-2, 0, 2$, respectively.
• Figure 3: Foliation on the pillowcase.
• Figure 4: Connected sum of the pillowcase with the 2-torus under construction (a).
• Figure 5: Connected sum of the pillowcase with the 2-torus under construction (b).

Theorems & Definitions (38)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Remark 2.2
• Remark 2.3
• Lemma 3.1
• proof
• Remark 3.2
• Lemma 3.3
• proof