The radiance obstruction and parallel forms on affine manifolds

Authors:
William Goldman and Morris W. Hirsch

Journal:
Trans. Amer. Math. Soc. **286** (1984), 629-649

MSC:
Primary 57R99; Secondary 53C20, 55R25

DOI:
https://doi.org/10.1090/S0002-9947-1984-0760977-7

MathSciNet review:
760977

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Abstract: A manifold is affine if it is endowed with a distinguished atlas whose coordinate changes are locally affine. When they are locally linear is called radiant. The obstruction to radiance is a one-dimensional class with coefficients in the flat tangent bundle of . Exterior powers of give information on the existence of parallel forms on , especially parallel volume forms. As applications, various kinds of restrictions are found on the holonomy and topology of compact affine manifolds.

**[**L. Auslander and L. Markus,**AM**]*Holonomy of flat affinely connected manifolds*, Ann. of Math. (2)**62**(1955), 139-159. MR**0072518 (17:298b)****[**G. Bredon,**Br**]*Sheaf theory*, McGraw-Hill, New York, 1967. MR**0221500 (36:4552)****[**W. G. Dwyer,**Dw**]*Vanishing cohomology over nilpotent groups*, Proc. Amer. Math. Soc.**49**(1975), 8-12. MR**0374242 (51:10442)****[**D. Fried and W. Goldman,**FG1**]*Three-dimensional affine crystallographic groups*, Adv. in Math.**47**(1983), 1-49. MR**689763 (84d:20047)****[**-, (in preparation).**FG2**]**[**D. Fried, W. Goldman and M. Hirsch,**FGH1**]*Affine manifolds and solvable groups*. Bull. Amer. Math. Soc.**3**(1980), 1045-1047. MR**585187 (81i:57018)****[**-,**FGH2**]*Affine manifolds with nilpotent holonomy*, Comment. Math. Helv.**56**(1981), 487-523. MR**656210 (83h:53062)****[**W. Goldman,**G**]*Discontinuous groups and the Euler class*, Doctoral Dissertation, Univ. of California, Berkeley, Calif., 1980.**[**W. Goldman and M. Hirsch,**GH1**]*A generalization of Bieberbach's theorem*, Invent. Math.**65**(1981), 1-11. MR**636876 (83f:53029)****[**-,**GH2**]*Polynomial forms on affine manifolds*, Pacific J. Math.**101**(1982), 115-121. MR**671843 (84f:53026)****[**-,**GH3**]*Affine structures and actions of Lie groups*(in preparation).**[**W. Goldman, M. Hirsch and G. Levitt,**GHL**]*Invariant measures for affine foliations*, Proc. Amer. Math. Soc.**86**(1982), 511-518. MR**671227 (84a:57026)****[**M. W. Hirsch,**H**]*Flat manifolds and the cohomology of groups*, Algebra and Geometric Topology, Lecture Notes in Math., vol. 664, Springer-Verlag, Berlin and New York, 1977. MR**518410 (80g:57057)****[**M. W. Hirsch and W. Thurston,**HT**]*Foliated bundles, flat manifolds and invariant measures*, Ann. of Math. (2)**101**(1975), 369-390. MR**0370615 (51:6842)****[**S. Kobayashi and K. Nomizu,**KN**]*Foundations of differential geometry*. Vol. I, Interscience, New York, 1963. MR**0152974 (27:2945)****[**G. A. Margulis,**Mg**]*Discrete groups of motions of spaces of nonpositive curvature*, Trans. Amer. Math. Soc.**109**(1977), 33-45.**[**J. W. Milnor,**Mi**]*On fundamental groups of complete affinely flat manifolds*, Adv. in Math.**25**(1977), 178-187. MR**0454886 (56:13130)****[**Y. Matsushima,**Mt**]*Affine structures on complex manifolds*, Osaka J. Math.**5**(1968), 215-222. MR**0240741 (39:2086)****[**L. Markus,**Mk**]*Cosmological models in differential geometry*, Mimeographed Notes, Univ. of Minnesota, 1962, p. 58.**[**J. P. Serre,**Se**]*Cohomologie des groupes discrets*, Prospects in Mathematics, Ann. of Math. Studies, no. 70, Princeton Univ. Press, Princeton, N. J., 1971, pp. 77-169. MR**0385006 (52:5876)****[**J. Smillie,**Sm1**]*Affinely flat manifolds*, Doctoral Dissertation, Univ. of Chicago, 1977.**[**-,**Sm2**]*An obstruction to the existence of affinely flat manifolds*, Invent. Math.**64**(1981), 411-415. MR**632981 (83a:53069)****[**R. J. Zimmer,**Z**]*Ergodic theory, group representations, and rigidity*, Bull. Amer. Math. Soc. (N.S.)**6**(1982), 383-416. MR**648527 (83k:22033)**

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DOI:
https://doi.org/10.1090/S0002-9947-1984-0760977-7

Article copyright:
© Copyright 1984
American Mathematical Society