Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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



Polynomials on affine manifolds

Author: David Fried
Journal: Trans. Amer. Math. Soc. 274 (1982), 709-719
MSC: Primary 53C15; Secondary 57R99, 58C05
MathSciNet review: 675076
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For a closed affine manifold $ M$ of dimension $ m$ the developing map defines an open subset $ D(\tilde M) \subset {{\mathbf{R}}^m}$. We show that $ D(\tilde M)$ cannot lie between parallel hyperplanes. When $ m \le 3$ we show that any nonconstant polynomial $ p:{{\mathbf{R}}^m} \to {\mathbf{R}}$ is unbounded on $ D(\tilde M)$. If $ D(\tilde M)$ lies in a half-space we show $ M$ has zero Euler characteristic. Under various special conditions on $ M$ we show that $ M$ has no nonconstant functions given by polynomials in affine coordinates.

References [Enhancements On Off] (What's this?)

  • [B] J. P. Benzecri, Variétés localement plates, Thesis, Princeton Univ., Princeton, N. J., 1955.
  • [Bo] A. Borel, Linear algebraic groups, Benjamin, New York, 1969. MR 0251042 (40:4273)
  • [FGH] D. Fried, W. Goldman and M. W. Hirsch, Affine manifolds with nilpotent holonomy, Comment. Math. Helv. 56 (1981), 487-523. MR 656210 (83h:53062)
  • [M] J. Milnor, On fundamental groups of complete affinely flat manifolds, Adv. in Math. 25 (1977), 178-187. MR 0454886 (56:13130)
  • [NY] T. Nagano and K. Yagi, The affine structures on the real $ 2$-torus. I, Osaka J. Math. 11 (1974), 181-210. MR 0377917 (51:14086)
  • [Th] W. Thurston, The geometry and topology of $ 3$-manifolds, Princeton Lecture Notes, Princeton Univ. Press, Princeton, N. J., Chapter IV, 1978.
  • [T] J. Tits, Free subgroups in linear groups, J. Algebra 20 (1972), 250-270. MR 0286898 (44:4105)
  • [W] J. Wolf, Spaces of constant curvature, McGraw-Hill, New York, 1967. MR 0217740 (36:829)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 53C15, 57R99, 58C05

Retrieve articles in all journals with MSC: 53C15, 57R99, 58C05

Additional Information

Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society