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ON COLLINEAR CLOSED ONE-FORMS

Part of: Differential topology

Published online by Cambridge University Press:  16 June 2011

IRINA GELBUKH
IRINA GELBUKH*
Affiliation:
Waseda Step 21, Room 403, Totsuka-Machi 1-103, Shinjuku-ku, Tokyo 169-0071, Japan (email: [email protected])
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Abstract

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We study one-forms with zero wedge-product, which we call collinear, and their foliations. We characterise the set of forms that define a given foliation, with special attention to closed forms and forms with small singular sets. We apply the notion of collinearity to give a criterion for the existence of a compact leaf and to study homological properties of compact leaves.

Keywords

differential one-formsingular setfoliationcompact leavescup-product

MSC classification

Secondary: 57R30: Foliations; geometric theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 84 , Issue 2 , October 2011 , pp. 322 - 336
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Arnoux, P. and Levitt, G., ‘Sur l’unique ergodicité des 1-formes fermées singulières’, Invent. Math. 84 (1986), 141–156.CrossRefGoogle Scholar
[2]Ayón-Beato, E., García, A., Macías, A. and Quevedo, H., ‘Static black holes of metric-affine gravity in the presence of matter’, Phys. Rev. D 64 (2001), 024026.CrossRefGoogle Scholar
[3]Bott, R. and Tu, L. W., Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, 82 (Springer, New York, 1982).CrossRefGoogle Scholar
[4]Farber, M., Topology of Closed One-forms, Mathematical Surveys, 108 (American Mathematical Society, Providence, RI, 2004).CrossRefGoogle Scholar
[5]Farber, M., Katz, G. and Levine, J., ‘Morse theory of harmonic forms’, Topology 37(3) (1998), 469–483.CrossRefGoogle Scholar
[6]Ferrando, J. J. and Sáez, J. A., ‘Type I vacuum solutions with aligned Papapetrou fields: an intrinsic charactrization’, J. Math. Phys. 47 (2006), 112501.CrossRefGoogle Scholar
[7]Gelbukh, I., ‘Presence of minimal components in a Morse form foliation’, Differential Geom. Appl. 22 (2005), 189–198.CrossRefGoogle Scholar
[8]Gelbukh, I., ‘Number of minimal components and homologically independent compact leaves for a Morse form foliation’, Studia Sci. Math. Hungar. 46(4) (2009), 547–557.Google Scholar
[9]Gelbukh, I., ‘On the structure of a Morse form foliation’, Czechoslovak Math. J. 59(1) (2009), 207–220.CrossRefGoogle Scholar
[10]Gelbukh, I., ‘Ranks of collinear Morse forms’, J. Geom. Phys. 61(2) (2011), 425–435.CrossRefGoogle Scholar
[11]Hehl, F. W. and Socorro, J., ‘Gauge theory of gravity: electrically charged solutions within the metric-affine framework’, Acta Phys. Polon. B 29 (1998), 1113–1120.Google Scholar
[12]Hurewicz, W. and Wallman, H., Dimension Theory (Princeton University Press, Princeton, NJ, 1996).Google Scholar
[13]Kahn, D. W., Introduction to Global Analysis, Pure and Applied Mathematics, 91 (Academic Press, New York, 1980).Google Scholar
[14]Mel’nikova, I., ‘A test for compactness of a foliation’, Math. Notes 58(6) (1995), 1302–1305.CrossRefGoogle Scholar
[15]Mel’nikova, I., ‘Singular points of a Morsian form and foliations’, Moscow Univ. Math. Bull. 51(4) (1996), 33–36.Google Scholar
[16]Mel’nikova, I., ‘Noncompact leaves of foliations of Morse forms’, Math. Notes 63(6) (1998), 760–763.CrossRefGoogle Scholar
[17]Mel’nikova, I., ‘Maximal isotropic subspaces of skew-symmetric bilinear mapping’, Moscow Univ. Math. Bull. 54(4) (1999), 1–3.Google Scholar
[18]Mulay, S. B., ‘Cycles and symmetries of zero-divisors’, Comm. Algebra 30 (2002), 3533–3558.CrossRefGoogle Scholar
[19]Sacksteder, R., ‘Foliations and pseudo-groups’, Amer. J. Math. 87 (1965), 79–102.CrossRefGoogle Scholar