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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Affine manifolds and orbits of algebraic groups


Authors: William M. Goldman and Morris W. Hirsch
Journal: Trans. Amer. Math. Soc. 295 (1986), 175-198
MSC: Primary 57R20; Secondary 32M10, 53C05, 55R25, 57R15
DOI: https://doi.org/10.1090/S0002-9947-1986-0831195-0
MathSciNet review: 831195
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Abstract: This paper is the sequel to The radiance obstruction and parallel forms on affine manifolds (Trans. Amer. Math. Soc. 286 (1984), 629-649) which introduced a new family of secondary characteristic classes for affine structures on manifolds. The present paper utilizes the representation of these classes in Lie algebra cohomology and algebraic group cohomology to deduce new results relating the geometric properties of a compact affine manifold $ {M^n}$ to the action on $ {{\mathbf{R}}^n}$ of the algebraic hull $ {\mathbf{A}}(\Gamma )$ of the affine holonomy group $ \Gamma \subseteq \operatorname{Aff}({{\mathbf{R}}^n})$.

A main technical result of the paper is that if $ M$ has a nonzero cohomology class represented by a parallel $ k$-form, then every orbit of $ {\mathbf{A}}(\Gamma )$ has dimension $ \geq k$. When $ M$ is compact, then $ {\mathbf{A}}(\Gamma )$ acts transitively provided that $ M$ is complete or has parallel volume; the converse holds when $ \Gamma $ is nilpotent. A $ 4$-dimensional subgroup of $ \operatorname{Aff}({{\mathbf{R}}^3})$ is exhibited which does not contain the holonomy group of any compact affine $ 3$-manifold.

When $ M$ has solvable holonomy and is complete, then $ M$ must have parallel volume. Conversely, if $ M$ has parallel volume and is of the homotopy type of a solvmanifold, then $ M$ is complete. If $ M$ is a compact homogeneous affine manifold or if $ M$ possesses a rational Riemannian metric, then it is shown that the conditions of parallel volume and completeness are equivalent.


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  • [1] L. Auslander, Simply transitive groups of affine motions, Amer. J. Math. 99 (1977), 809-821. MR 0447470 (56:5782)
  • [2] J. P. Benzecri, Variétés localement affines, Thèse, Princeton Univ., Princeton, N.J., 1955.
  • [3] -, Variétés localement affines et projectives, Bull. Soc. Math. France 88 (1960), 229-332. MR 0124005 (23:A1325)
  • [4] N. Boyom, Structures affines sur les groupes de Lie nilopotents (preprint).
  • [5] G. Bredon, Sheaf theory, McGraw-Hill, New York, 1967. MR 0221500 (36:4552)
  • [6] H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, N.J., 1956. MR 0077480 (17:1040e)
  • [7] C. Chevalley and S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948), 85-124. MR 0024908 (9:567a)
  • [8] B. Evans and L. Moser, Solvable fundamental groups of compact $ 3$-manifolds, Trans. Amer. Math. Soc. 168 (1972), 189-210. MR 0301742 (46:897)
  • [9] D. Fried, Polynomial on affine manifolds, Trans. Amer. Math. Soc. 274 (1982), 709-719. MR 675076 (84d:53035)
  • [10] -, Distality, completeness, and affine structures, J. Differential Geom. (to appear). MR 868973 (88i:53060)
  • [11] D. Fried and W. Goldman, Three-dimensional affine crystallographic groups, Adv. in Math. 47 (1983), 1-49. MR 689763 (84d:20047)
  • [12] D. Fried, W. Goldman and M. W. Hirsch, Affine manifolds and solvable groups, Bull. Amer. Math. Soc. (N.S.) 3 (1980), 1045-1047. MR 585187 (81i:57018)
  • [13] -, Affine manifolds with nilpotent holonomy, Comment. Math. Helv. 56 (1981), 487-523. MR 656210 (83h:53062)
  • [14] W. Goldman, Affine manifolds and projective geometry on surfaces, Senior Thesis, Princeton Univ., Princeton, N.J., 1977.
  • [15] -, Discontinuous groups and the Euler class, Doctoral Dissertation, Univ. of California, Berkeley, 1980.
  • [16] -, Two examples of affine manifolds, Pacific J. Math. 94 (1981), 327-330. MR 628585 (83a:57028)
  • [17] -, On the polynomial cohomology of affine manifolds, Invent. Math 65 (1982), 453-457. MR 643563 (83d:53048)
  • [18] -, Projective structures with Fuchsian holonomy (submitted).
  • [19] -, (in preparation).
  • [20] W. Goldman and M. W. Hirsch, A generalization of Bieberbach's theorem, Invent. Math. 65 (1981), 1-11. MR 636876 (83f:53029)
  • [21] -, Polynomial forms on affine manifolds, Pacific J. Math 101 (1982), 115-121. MR 671843 (84f:53026)
  • [22] -, The radiance obstruction and parallel forms on affine manifolds, Trans. Amer. Math. Soc. 286 (1984), 629-649. MR 760977 (86f:57032)
  • [23] W. Goldman, M. W. Hirsch and G. Levitt, Invariant measures for affine foliations, Proc. Amer. Math. Soc. 86 (1982), 511-518. MR 671227 (84a:57026)
  • [24] A. Haefliger, Differentiable cohomology, C.I.M.E. lectures, Varenna, Italy, 1979.
  • [25] J. Helmstetter, Algèbres symétriques à gauche, C. R. Acad. Sci. Paris 272 (1971), 1088-1091. MR 0291233 (45:327)
  • [26] -, Radical d'une algèbre symétrique à gauche, Ann. Inst. Fourier (Grenoble) 29 (1979), 17-35.
  • [27] M. W. Hirsch, Flat manifolds and the cohomology of groups, Algebraic and Geometric Topology, Lecture Notes in Math., vol. 664, Springer-Verlag, New York, 1977. MR 518410 (80g:57057)
  • [28] M. W. Hirsch and W. P. Thurston, Foliated bundles, flat manifolds, and invariant measures, Ann. of Math. 101 (1975), 369-390. MR 0370615 (51:6842)
  • [29] G. Hochschild, Cohomology of algebraic linear groups, Illinois J. Math. 5 (1961), 492-519. MR 0130901 (24:A755)
  • [30] -, Basic theory of algebraic groups and Lie algebras, Graduate Texts in Math., vol. 75, Springer-Verlag, Berlin, Heidelberg and New York, 1981. MR 620024 (82i:20002)
  • [31] J. Humphreys, Linear algebraic groups, Graduate Texts in Math., vol. 21, Springer-Verlag, Berlin, Heidelberg and New York, 1975. MR 0396773 (53:633)
  • [32] V. Kac and E. B. Vinberg, Quasi-homogeneous cones, Math. Notes 1 (1967), 231-235 (translated from Math. Zametki 1 (1967), 347-354). MR 0208470 (34:8280)
  • [33] J. L. Koszul, Homologie et cohomologie des algèbres de Lie, Bull. Soc. Math. France 78 (1950), 66-127. MR 0036511 (12:120g)
  • [34] -, Domaines bornés homogènes et orbites des groupes de transformations affine, Bull. Soc. Math. France 89 (1961), 515-533. MR 0145559 (26:3090)
  • [35] -, Ouvertes convexes homogènes des espaces affine, Math. Z. 139 (1974), 254-259.
  • [36] L. Markus, Cosmological models in differential geometry, mimeographed notes, Univ. of Minnesota, 1962, p. 58.
  • [37] Y. Matsushima, Affine structures on complex manifolds, Osaka J. Math. 5 (1968), 215-222. MR 0240741 (39:2086)
  • [38] A. Medina Perea, Flat left-invariant connections adapted to the automorphism structure of a Lie group, J. Differential Geometry 16 (1981), 445-474. MR 654637 (83j:53023)
  • [39] -, (to appear).
  • [40] J. W. Milnor, On fundamental groups of complete affinely flat manifolds, Adv. in Math. 25 (1977), 178-187. MR 0454886 (56:13130)
  • [41] G. D. Mostow, Cohomology of topological groups and solvmanifolds, Ann. of Math. 73 (1961), 20-48. MR 0125179 (23:A2484)
  • [42] M. S. Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Math., Band 68, Springer-Verlag, Berlin, Heidelberg and New York, 1972. MR 0507234 (58:22394a)
  • [43] M. Rosenlicht, On quotient varieties and the embedding of certain homogeneous spaces, Trans. Amer. Math. Soc. 101 (1961), 211-233. MR 0130878 (24:A732)
  • [44] H. Shima, Homogeneous Hessian manifolds, Manifolds and Lie Groups, Papers in honor of Yozo Matsushima, Progress in Math., vol. 14, Birkhäuser, Boston, Mass., 1981, pp. 385-392. MR 642868 (83h:53066)
  • [45] J. Smillie, Affinely flat manifolds, Doctoral Dissertation, Univ. of Chicago, 1977.
  • [46] -, An obstruction to the existence of affinely flat manifolds, Invent. Math. 64 (1981), 411-415. MR 632981 (83a:53069)
  • [47] -, Affine structures with diagonal holonomy, I.A.S. preprint (1979).
  • [48] -, Complex affine manifolds with nilpotent holonomy (in preparation).
  • [49] D. Sullivan and W. Thurston, Manifolds with canonical coordinates: some examples, Enseign. Math. 29 (1983), 15-25. MR 702731 (84i:53035)
  • [50] J. Vey, Sur une notion d'hyperbolicité des variétés localement plates, These, Univ. de Grenoble, 1969.
  • [51] -, Sur une notion d'hyperbolicité des variétés localement plates, C. R. Acad. Sci. Paris 266 (1968), 622-624.
  • [52] -, Sur les automorphismes affines des ouverts convexes saillants, Ann. Scuola Norm. Sup. Pisa 4 (1970), 641-665. MR 0283720 (44:950)
  • [53] E. B. Vinberg, Homogeneous convex cones, Trans. Moscow Math. Soc. 12 (1963), 340-403. MR 0158414 (28:1637)
  • [54] K. Yagi, Hessian structures on affine manifolds, Manifolds and Lie Groups, Papers in honor of Yozo Matsushima, Progress in Math., vol. 14, Birkhäuser, Boston, Mass., 1981. MR 642872 (83h:53067)
  • [55] -, On compact homogeneous affine manifolds, Osaka J. Math. 7 (1970), 457-475. MR 0284943 (44:2167)
  • [56] H. Kim, On complete left-invariant affine structures on $ 4$-dimensional nilpotent Lie groups, Thesis, Univ. of Michigan, 1983.
  • [57] H. Shima, On certain locally flat homogeneous manifolds of solvable Lie groups, Osaka J. Math. 13 (1976), 213-229. MR 0413010 (54:1131)
  • [58] H. Hironaka, Triangulations of algebraic sets, Algebraic Geometry, Arcata 1974, Proc. Sympos. Pure Math., vol. 29, Amer. Math. Soc., Providence, R.I., 1975, pp. 165-186. MR 0374131 (51:10331)

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DOI: https://doi.org/10.1090/S0002-9947-1986-0831195-0
Article copyright: © Copyright 1986 American Mathematical Society

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