Let $X$ be a connected non-compact $2$-dimensional manifold possibly with boundary and $\Delta$ be a foliation on $X$ such that each leaf $\omega\in\Delta$ is homeomorphic to $\mathbb{R}$ and has a trivially foliated neighborhood. Such foliations on the plane were studied by W. Kaplan who also gave their topological classification. He proved that the plane splits into a family of open strips foliated by parallel lines and glued along some boundary intervals. However W. Kaplan's construction depends on a choice of those intervals, and a foliation is described in a non-unique way. We propose a canonical cutting by open strips which gives a uniqueness of classifying invariant. We also describe topological types of closures of those strips under additional assumptions on $\Delta$.

Sergiy Maksymenko and Eugene Polulyakh, Foliations with all non-closed leaves on non-compact surfaces, Methods Funct. Anal. Topology 22
(2016), no. 3, 266-282.

BibTex

@article {MFAT884,
AUTHOR = {Maksymenko, Sergiy and Polulyakh, Eugene},
TITLE = {Foliations with all non-closed leaves on non-compact surfaces},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {22},
YEAR = {2016},
NUMBER = {3},
PAGES = {266-282},
ISSN = {1029-3531},
URL = {http://mfat.imath.kiev.ua/article/?id=884},
}

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