Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-20T15:23:47.889Z Has data issue: false hasContentIssue false
  • English
  • Français

Closure of Leaves in Transversely Affine Foliations

Published online by Cambridge University Press:  20 November 2018

Robert A. Wolak
Robert A. Wolak*
Affiliation:
Departamento de Xeometria e Topoloxia, Facultade de Matematicas, Universidade de Santiago de Compostela, 15705 Santiago de Compostela, Spain
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We present first examples of complete transversely affine foliations on compact manifolds with leaves whose closures are not submanifolds. Moreover, we prove that under some additional assumptions the closures of leaves form a singular foliation.

Keywords

57R30
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 34 , Issue 4 , 01 December 1991 , pp. 553 - 558
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Banchoff, T. F., Rosen, M. I., Periodic points of Anosov diffeomorphisms, Global Analysis, Proc. Symp. Pure Math. 14 (1970), 17–21.Google Scholar
2. Blumenthal, R. A., Hebda, J. J., Ehresmann connections for foliations, Indiana Un. Math. J. (4)33 (1984), 597–611.Google Scholar
3. Gantmacher, F. R., The theory of matrices, vol. 1, 2. Chelsea Publ., New York, 1959.Google Scholar
4. Ghys, E., Un feuilletage analytique dont la cohomologie basique est de dimension infinie, Publ. IRMA Lille (1)VII(1985).Google Scholar
5. Glimm, J., Locally compact transformation groups, Trans. A.M.S. 101 (1961), 124–138.Google Scholar
6. Goldman, W., Hirsch, M., Levitt, G., Invariant measures for affine foliations, Proc. A.M.S. 86(1982), 511– 518.Google Scholar
7. Hancock, S. G., Construction of invariant sets for Anosov diffeomorphisms, J. London Math. Soc. 18 (1978), 339–348.Google Scholar
8. Hirsch, M. W., On invariant sets of Anosov diffeomorphisms. Essays on Topology and Related Topics, eds. Haefligerand, A. Narasimhan, R., Springer, 1970, 126–135.Google Scholar
9. R. Mané, Invariant sets of Anosov diffeomorphisms, Inventiones Math. 46 (1978), 147–152.Google Scholar
10. Molino, P., Feuilletages riemanniens sur les variétés compactes; champs de Killing transverses, C.R. Acad. Sci. Paris 289 (1979), 421–423.Google Scholar
11. Molino, P., Géométrie globale des feuilletages riemanniens, Proc. Kon. Nederl. Acad. 85 (1982), 45–76.Google Scholar
12. Molino, P., Riemannian foliations, Progress in Math. 73, Birkhâuser, 1988.Google Scholar
13. Moore, C. C., Distal affine transformation groups, Amer. J. Math. 90 (1968), 733–751.Google Scholar
14. Plante, J., Anosow flows, transversely affine foliations and conjecture of Verjovsky, J. London Math. Soc. 23 (1981), 359–362.Google Scholar
15. Porteous, H. L., Anosov diffeomorphisms of flat manifolds, Topology 11 (1972), 307–315.Google Scholar
16. Przytycki, F., Construction of invariant sets for Anosov diffeomorphisms and hyperbolic attractors, Studia Math. 68 (1980), 199–213.Google Scholar
17. Reinhart, B., Foliated manifolds with bundle-like metrics, Ann. of Math. 69 (1959), 119–132.Google Scholar
18. Rosenlicht, M., On quotient varieties and the affine embedding of certain homogeneous spaces, Trans. A.M.S. 101 (1961), 211–223.Google Scholar
19. Smale, S., Differentiable dynamical systems, Bull. A.M.S. 73 (1967), 747–817.Google Scholar
20. Stefan, P., Accessible sets, orbits and foliations with singularities, Proc. London Math. Soc. 29(1974), 699– 713.Google Scholar
21. Sussmann, H., Orbits of vector fields and integrability of distributions, Trans. A.M.S. 180 (1973), 171–188.Google Scholar
22. Tondeur, Ph., Foliations on Riemannian manifolds. Springer-Verlag, 1988.Google Scholar
23. Winkelnkemper, H., The graph of a foliation, Ann. Glob. Anal. Geom. 1 (1983), 51–75.Google Scholar
24. Wolak, R. A., Foliations admitting transverse systems of differential equations, Comp. Math. 67 (1988), 89–101.Google Scholar
25. Wolak, R. A., Le graphe d'un feuilletage admettant un système d'équations différentielles transverse, Math. Z. 201 (1989), 177–182.Google Scholar
26. Wolak, R. A., Transversely affine foliations compared with affine manifolds, Quart. J. Math (Oxford) 41 (1990), 369–384.Google Scholar
27. Wolak, R. A., Foliated G-structures and Riemannian foliations, Manuscripta Math. 66 (1989), 45–59.Google Scholar