×

zbMATH — the first resource for mathematics

The number of minimal components and homologically independent compact leaves of a weakly generic Morse form on a closed surface. (English) Zbl 1280.57021
Let \(M^2_g\) be a closed orientable surface of genus \(g\), and consider the foliation induced on \(M\) by a weakly generic Morse form \(\omega\). The author relates three quantities: the number of homologically independent compact leaves of the foliation, the number of its minimal components, and the total number of singularities of \(\omega\) that are surrounded by a minimal component.

MSC:
57R30 Foliations in differential topology; geometric theory
58K65 Topological invariants on manifolds
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] Pierre Arnoux and Gilbert Levitt, Sur l’unique ergodicité des 1-formes fermées singulières , Invent. Math. 84 (1986), 141-156. · Zbl 0577.58021 · doi:10.1007/BF01388736 · eudml:143337
[2] Michael Farber, Topology of closed one-forms , Math. Surv. 108 , American Mathematical Society, 2004. · Zbl 1052.58016
[3] Irina Gelbukh, Presence of minimal components in a Morse form foliation , Diff. Geom. Appl. 22 (2005), 189-198. · Zbl 1070.57016 · doi:10.1016/j.difgeo.2004.10.006
[4] —, Number of minimal components and homologically independent compact leaves for a Morse form foliation , Stud. Sci. Math. Hung. 46 (2009), 547-557. · Zbl 1274.57005 · doi:10.1556/SScMath.2009.1108
[5] —, On the structure of a Morse form foliation , Czech. Math. J. 59 (2009), 207-220. · Zbl 1224.57010 · doi:10.1007/s10587-009-0015-5
[6] —, Structure of a Morse form foliation on a closed surface in terms of genus , Diff. Geom. Appl. 29 (2011), 473-492. · Zbl 1223.57022 · doi:10.1016/j.difgeo.2011.04.029
[7] Frank Harary, Graph theory , Addison-Wesley Publishing Company, Massachusetts, 1994.
[8] Hideki Imanishi, On codimension one foliations defined by closed one forms with singularities , J. Math. Kyoto Univ. 19 (1979), 285-291. · Zbl 0417.57010
[9] Anatole Katok, Invariant measures for flows on oriented surfaces , Sov. Math. Dokl. 14 (1973), 1104-1108. · Zbl 0298.28013
[10] Artemiy Grigorievich Maier, Trajectories on closed orientable surfaces , Math. Sbor. 12 (1943), 71-84. · Zbl 0063.03856
[11] Irina Mel’nikova, An indicator of the noncompactness of a foliation on \(M^2_g\) , Math. Notes 53 (1993), 356-358. · Zbl 0809.57018 · doi:10.1007/BF01207728
[12] —, A test for non-compactness of the foliation of a Morse form , Russ. Math. Surv. 50 (1995), 444-445. · Zbl 0859.58005 · doi:10.1070/RM1995v050n02ABEH002092
[13] —, Maximal isotropic subspaces of skew-symmetric bilinear mapping , Moscow Univ. Math. Bull. 54 (1999), 1-3. \noindentstyle · Zbl 0957.57018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.