Holonomy and metric properties of foliations in higher codimension
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- by Alexander Morgan PDF
- Proc. Amer. Math. Soc. 58 (1976), 255-261 Request permission
Abstract:
It is well known that a codimension 1 foliation with finite holonomy on a compact manifold must have a bundle-like metric. A counterexample is presented to the higher codimension generalization of this theorem. However, a stronger holonomy restriction (expressed via the Bott connection) is shown to imply the existence of a bundle-like metric.References
- Raoul Bott, Lectures on characteristic classes and foliations, Lectures on algebraic and differential topology (Second Latin American School in Math., Mexico City, 1971) Lecture Notes in Math., Vol. 279, Springer, Berlin, 1972, pp. 1–94. Notes by Lawrence Conlon, with two appendices by J. Stasheff. MR 0362335
- H. Furstenberg, The structure of distal flows, Amer. J. Math. 85 (1963), 477–515. MR 157368, DOI 10.2307/2373137
- André Haefliger, Variétés feuilletées, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 16 (1962), 367–397 (French). MR 189060
- André Haefliger, Homotopy and integrability, Manifolds–Amsterdam 1970 (Proc. Nuffic Summer School), Lecture Notes in Mathematics, Vol. 197, Springer, Berlin, 1971, pp. 133–163. MR 0285027
- H. Blaine Lawson Jr., Foliations, Bull. Amer. Math. Soc. 80 (1974), 369–418. MR 343289, DOI 10.1090/S0002-9904-1974-13432-4
- J. S. Pasternack, Foliations and compact Lie group actions, Comment. Math. Helv. 46 (1971), 467–477. MR 300307, DOI 10.1007/BF02566859
- Bruce L. Reinhart, Foliated manifolds with bundle-like metrics, Ann. of Math. (2) 69 (1959), 119–132. MR 107279, DOI 10.2307/1970097
- Richard Sacksteder, Foliations and pseudogroups, Amer. J. Math. 87 (1965), 79–102. MR 174061, DOI 10.2307/2373226
- Norman Steenrod, The Topology of Fibre Bundles, Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, N. J., 1951. MR 0039258 W. Thurston, Foliations of $3$-manifolds which are circle bundles, Thesis, Univ. of California, Berkeley, Calif., 1972.
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 58 (1976), 255-261
- MSC: Primary 57D30
- DOI: https://doi.org/10.1090/S0002-9939-1976-0410761-X
- MathSciNet review: 0410761