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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.

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[1]Arnoux, P. and Levitt, G., ‘Sur l’unique ergodicité des 1-formes fermées singulières’, Invent. Math. 84 (1986), 141156.
[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.
[3]Bott, R. and Tu, L. W., Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, 82 (Springer, New York, 1982).
[4]Farber, M., Topology of Closed One-forms, Mathematical Surveys, 108 (American Mathematical Society, Providence, RI, 2004).
[5]Farber, M., Katz, G. and Levine, J., ‘Morse theory of harmonic forms’, Topology 37(3) (1998), 469483.
[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.
[7]Gelbukh, I., ‘Presence of minimal components in a Morse form foliation’, Differential Geom. Appl. 22 (2005), 189198.
[8]Gelbukh, I., ‘Number of minimal components and homologically independent compact leaves for a Morse form foliation’, Studia Sci. Math. Hungar. 46(4) (2009), 547557.
[9]Gelbukh, I., ‘On the structure of a Morse form foliation’, Czechoslovak Math. J. 59(1) (2009), 207220.
[10]Gelbukh, I., ‘Ranks of collinear Morse forms’, J. Geom. Phys. 61(2) (2011), 425435.
[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), 11131120.
[12]Hurewicz, W. and Wallman, H., Dimension Theory (Princeton University Press, Princeton, NJ, 1996).
[13]Kahn, D. W., Introduction to Global Analysis, Pure and Applied Mathematics, 91 (Academic Press, New York, 1980).
[14]Mel’nikova, I., ‘A test for compactness of a foliation’, Math. Notes 58(6) (1995), 13021305.
[15]Mel’nikova, I., ‘Singular points of a Morsian form and foliations’, Moscow Univ. Math. Bull. 51(4) (1996), 3336.
[16]Mel’nikova, I., ‘Noncompact leaves of foliations of Morse forms’, Math. Notes 63(6) (1998), 760763.
[17]Mel’nikova, I., ‘Maximal isotropic subspaces of skew-symmetric bilinear mapping’, Moscow Univ. Math. Bull. 54(4) (1999), 13.
[18]Mulay, S. B., ‘Cycles and symmetries of zero-divisors’, Comm. Algebra 30 (2002), 35333558.
[19]Sacksteder, R., ‘Foliations and pseudo-groups’, Amer. J. Math. 87 (1965), 79102.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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