Invariant measures for affine foliations. (English) Zbl 0534.57014
The authors consider foliations with an affine transverse structure. The main result is that if such a foliation has a nilpotent affine homology group then it admits a nontrivial transverse measure. In particular, this implies that a certain cohomology class of a compact affine manifold with nilpotent holonomy group is nonzero.
Reviewer: S.Goodman
MSC:
| 57R30 | Foliations in differential topology; geometric theory |
| 53C12 | Foliations (differential geometric aspects) |
| 28D15 | General groups of measure-preserving transformations |
| 53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
References:
| [1] | Peter Furness and Edmond Fedida, Feuilletages transversalement affines de codimension 1, C. R. Acad. Sci. Paris Sér. A-B 282 (1976), no. 16, Ai, A825 – A827 (French, with English summary). · Zbl 0321.57017 |
| [2] | David Fried, William Goldman, and Morris W. Hirsch, Affine manifolds with nilpotent holonomy, Comment. Math. Helv. 56 (1981), no. 4, 487 – 523. · Zbl 0516.57014 · doi:10.1007/BF02566225 |
| [3] | P. M. D. Furness and E. Fédida, Tranversally affine foliations, Glasgow Math. J. 17 (1976), no. 2, 106 – 111. · Zbl 0329.57004 · doi:10.1017/S0017089500002810 |
| [4] | W. Goldman, Discontinuous groups and the Euler class, §1, Ph.D. Thesis, University of California, Berkeley, 1980. |
| [5] | William M. Goldman and Morris W. Hirsch, Polynomial forms on affine manifolds, Pacific J. Math. 101 (1982), no. 1, 115 – 121. · Zbl 0522.57022 |
| [6] | -, Parallel characteristic class of affine manifolds, (in preparation). |
| [7] | W. Gottschalk and G. Hedlund, Topological dyamics, Amer. Math. Soc. Colloq. Publ., vol. 36, Amer. Math. Soc., Providence, R.I., 1965. |
| [8] | Dennis A. Hejhal, Monodromy groups and linearly polymorphic functions, Acta Math. 135 (1975), no. 1, 1 – 55. · Zbl 0333.34002 · doi:10.1007/BF02392015 |
| [9] | R. S. Kulkarni, On the principle of uniformization, J. Differential Geom. 13 (1978), no. 1, 109 – 138. · Zbl 0381.53023 |
| [10] | J. F. Plante, Foliations with measure preserving holonomy, Ann. of Math. (2) 102 (1975), no. 2, 327 – 361. · Zbl 0314.57018 · doi:10.2307/1971034 |
| [11] | Masayoshi Miyanishi, On group actions, Recent progress of algebraic geometry in Japan, North-Holland Math. Stud., vol. 73, North-Holland, Amsterdam, 1983, pp. 152 – 187. · Zbl 0529.14026 · doi:10.1016/S0304-0208(09)70009-X |
| [12] | David Ruelle and Dennis Sullivan, Currents, flows and diffeomorphisms, Topology 14 (1975), no. 4, 319 – 327. · Zbl 0321.58019 · doi:10.1016/0040-9383(75)90016-6 |
| [13] | Bobo Seke, Sur les structures transversalement affines des feuilletages de codimension un, Ann. Inst. Fourier (Grenoble) 30 (1980), no. 1, ix, 1 – 29 (French, with English summary). · Zbl 0417.57011 |
| [14] | D. Sullivan and W. Thurston, Manifolds with canonical coordinates: some examples, Inst. Hautes Etude Sci. Publ. Math., 1979 (preprint). · Zbl 0529.53025 |
| [15] | William P. Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, vol. 35, Princeton University Press, Princeton, NJ, 1997. Edited by Silvio Levy. · Zbl 0873.57001 |
| [16] | Robert A. Blumenthal, Transversely homogeneous foliations, Ann. Inst. Fourier (Grenoble) 29 (1979), no. 4, vii, 143 – 158 (English, with French summary). · Zbl 0405.57016 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
