Holonomy and metric properties of foliations in higher codimension

Morgan, Alexander

doi:10.1090/S0002-9939-1976-0410761-X

Holonomy and metric properties of foliations in higher codimension
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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

Foliations of $3$-manifolds which are circle bundles