Open Access

# Foliations with all non-closed leaves on non-compact surfaces

### Abstract

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 $\Delta$.

Key words: Foliation, non-compact surface, fiber bundles.

### Article Information

 Title Foliations with all non-closed leaves on non-compact surfaces Source Methods Funct. Anal. Topology, Vol. 22 (2016), no. 3, 266-282 MathSciNet MR3554653 zbMATH 06742111 Milestones Received 30/05/2016 Copyright The Author(s) 2016 (CC BY-SA)

### Authors Information

Sergiy Maksymenko
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Teresh\-chenkivs'ka, Kyiv, 01601, Ukraine

Eugene Polulyakh
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Teresh\-chenkivs'ka, Kyiv, 01601, Ukraine

### Citation Example

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},
MRNUMBER = {MR3554653},
ZBLNUMBER = {06742111},
URL = {http://mfat.imath.kiev.ua/article/?id=884},
}

### References

1. A. Andronov and Pontryagin L., Systèmes grossiers, Dokl. Akad. Nauk. SSSR 14 (1937), 247-251.
2. S. H. Aranson and V. Z. Grines, On some invariants of dynamical systems on two-dimensional manifolds (necessary and sufficient conditions for the topological equivalence of transitive dynamical systems), Math. USSR Sb. 19 (1973), no. 3, 365-394. CrossRef
3. Kh. S. Aranson and V. Z. Grines, Topological classification of flows on closed two-dimensional manifolds, Russ. Math. Surv. 41 (1986), no. 1, 183-208. CrossRef
4. Kh. S. Aranson, V. Z. Grines, and V. A. Kaimanovich, Classification of supertransitive 2-webs on surfaces, J. Dynam. Control Systems 9 (2003), no. 4, 455-468.  MathSciNet CrossRef
5. Kh. S. Aranson, E. V. Zhuzhoma, and V. S. Medvedev, On continuity of geodesic frameworks of flows on surfaces, Sb. Math. 188 (1997), no. 7, 955-972.  MathSciNet CrossRef
6. William M. Boothby, The topology of regular curve families with multiple saddle points, Amer. J. Math. 73 (1951), 405-438.  MathSciNet
7. William M. Boothby, The topology of the level curves of harmonic functions with critical points, Amer. J. Math. 73 (1951), 512-538.  MathSciNet
8. Idel Bronstein and Igor Nikolaev, Peixoto graphs of Morse-Smale foliations on surfaces, Topology Appl. 77 (1997), no. 1, 19-36.  MathSciNet CrossRef
9. N. V. Budnytska and O. O. Pryshlyak, Equivalence of closed 1-forms on surfaces with edge, Ukrainian Math. J. 61 (2009), no. 11, 1710-1727.  MathSciNet CrossRef
10. N. V. Budnytska and T. V. Rybalkina, Realization of a closed 1-form on closed oriented surfaces, Ukrainian Math. J. 64 (2012), no. 6, 844-856.  MathSciNet CrossRef
11. Michael Farber, Topology of closed one-forms, Mathematical Surveys and Monographs, vol. 108, American Mathematical Society, Providence, RI, 2004.  MathSciNet CrossRef
12. James A. Jenkins and Marston Morse, Contour equivalent pseudoharmonic functions and pseudoconjugates, Amer. J. Math. 74 (1952), 23-51.  MathSciNet
13. Wilfred Kaplan, Regular curve-families filling the plane, I, Duke Math. J. 7 (1940), 154-185.  MathSciNet
14. Sergey Maksymenko, Stabilizers and orbits of smooth functions, Bull. Sci. Math. 130 (2006), no. 4, 279-311.  MathSciNet CrossRef
15. Sergiy Maksymenko and Eugene Polulyakh, Foliations with non-compact leaves on surfaces, Proc. Intern. Geom. Center 8 (2015), no. 3--4, 17-30.
16. M. Morse, The existence of pseudoconjugates on Riemann surfaces, Fund. Math. 39 (1952), 269-287 (1953).  MathSciNet
17. A. A. Oshemkov and V. V. Sharko, Classification of Morse-Smale flows on two-dimensional manifolds, Sb. Math. 189 (1998), no. 8, 1205–1250.  MathSciNet CrossRef
18. M. M. Peixoto, Structural stability on two-dimensional manifolds, Topology 1 (1962), 101-120.  MathSciNet
19. M. M. Peixoto, Structural stability on two-dimensional manifolds. A further remark., Topology 2 (1963), 179-180.  MathSciNet
20. L. P. Plachta, The combinatorics of gradient-like flows and foliations on closed surfaces. II. The problem of realization and some estimates, Mat. Metodi Fiz.-Mekh. Polya 44 (2001), no. 2, 7-16.  MathSciNet
21. L. P. Plachta, The combinatorics of gradient-like flows and foliations on closed surfaces. III. The problem of realization and some estimates, Mat. Metodi Fiz.-Mekh. Polya 44 (2001), no. 3, 7-16.  MathSciNet
22. Leonid Plachta, The combinatorics of gradient-like flows and foliations on closed surfaces. I. Topological classification, Topology Appl. 128 (2003), no. 1, 63-91.  MathSciNet CrossRef