The radiance obstruction and parallel forms on affine manifolds

Goldman, William; Hirsch, Morris W.

doi:10.1090/S0002-9947-1984-0760977-7

The radiance obstruction and parallel forms on affine manifolds
HTML articles powered by AMS MathViewer

by William Goldman and Morris W. Hirsch

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

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

PDF | Request permission

Abstract:

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

References

L. Auslander and L. Markus, Holonomy of flat affinely connected manifolds, Ann. of Math. (2) 62 (1955), 139–151. MR 72518, DOI 10.2307/2007104
Glen E. Bredon, Sheaf theory, McGraw-Hill Book Co., New York-Toronto-London, 1967. MR 0221500
William G. Dwyer, Vanishing homology over nilpotent groups, Proc. Amer. Math. Soc. 49 (1975), 8–12. MR 374242, DOI 10.1090/S0002-9939-1975-0374242-3
David Fried and William M. Goldman, Three-dimensional affine crystallographic groups, Adv. in Math. 47 (1983), no. 1, 1–49. MR 689763, DOI 10.1016/0001-8708(83)90053-1
D. Fried, W. Goldman, and M. W. Hirsch, Affine manifolds and solvable groups, Bull. Amer. Math. Soc. (N.S.) 3 (1980), no. 3, 1045–1047. MR 585187, DOI 10.1090/S0273-0979-1980-14849-1
David Fried, William Goldman, and Morris W. Hirsch, Affine manifolds with nilpotent holonomy, Comment. Math. Helv. 56 (1981), no. 4, 487–523. MR 656210, DOI 10.1007/BF02566225

Discontinuous groups and the Euler class

William M. Goldman and Morris W. Hirsch, A generalization of Bieberbach’s theorem, Invent. Math. 65 (1981/82), no. 1, 1–11. MR 636876, DOI 10.1007/BF01389291
William M. Goldman and Morris W. Hirsch, Polynomial forms on affine manifolds, Pacific J. Math. 101 (1982), no. 1, 115–121. MR 671843

Affine structures and actions of Lie groups

William M. Goldman, Morris W. Hirsch, and Gilbert Levitt, Invariant measures for affine foliations, Proc. Amer. Math. Soc. 86 (1982), no. 3, 511–518. MR 671227, DOI 10.1090/S0002-9939-1982-0671227-8
Morris W. Hirsch, Flat manifolds and the cohomology of groups, Algebraic and geometric topology (Proc. Sympos., Univ. California, Santa Barbara, Calif., 1977) Lecture Notes in Math., vol. 664, Springer, Berlin, 1978, pp. 94–103. MR 518410
Morris W. Hirsch and William P. Thurston, Foliated bundles, invariant measures and flat manifolds, Ann. of Math. (2) 101 (1975), 369–390. MR 370615, DOI 10.2307/1970996
Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1963. MR 0152974

Discrete groups of motions of spaces of nonpositive curvature

109

John Milnor, On fundamental groups of complete affinely flat manifolds, Advances in Math. 25 (1977), no. 2, 178–187. MR 454886, DOI 10.1016/0001-8708(77)90004-4
Yozô Matsushima, Affine structures on complex manifolds, Osaka Math. J. 5 (1968), 215–222. MR 240741

Cosmological models in differential geometry

Jean-Pierre Serre, Cohomologie des groupes discrets, Prospects in mathematics (Proc. Sympos., Princeton Univ., Princeton, N.J., 1970) Ann. of Math. Studies, No. 70, Princeton Univ. Press, Princeton, N.J., 1971, pp. 77–169 (French). MR 0385006

Affinely flat manifolds

John Smillie, An obstruction to the existence of affine structures, Invent. Math. 64 (1981), no. 3, 411–415. MR 632981, DOI 10.1007/BF01389273
Robert J. Zimmer, Ergodic theory, group representations, and rigidity, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 383–416. MR 648527, DOI 10.1090/S0273-0979-1982-15005-4