Polynomials on affine manifolds
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- by David Fried
- Trans. Amer. Math. Soc. 274 (1982), 709-719
- DOI: https://doi.org/10.1090/S0002-9947-1982-0675076-0
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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
- J. P. Benzecri, Variétés localement plates, Thesis, Princeton Univ., Princeton, N. J., 1955.
- Armand Borel, Linear algebraic groups, W. A. Benjamin, Inc., New York-Amsterdam, 1969. Notes taken by Hyman Bass. MR 0251042
- 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
- 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
- Tadashi Nagano and Katsumi Yagi, The affine structures on the real two-torus. I, Osaka Math. J. 11 (1974), 181–210. MR 377917 W. Thurston, The geometry and topology of $3$-manifolds, Princeton Lecture Notes, Princeton Univ. Press, Princeton, N. J., Chapter IV, 1978.
- J. Tits, Free subgroups in linear groups, J. Algebra 20 (1972), 250–270. MR 286898, DOI 10.1016/0021-8693(72)90058-0
- Joseph A. Wolf, Spaces of constant curvature, McGraw-Hill Book Co., New York-London-Sydney, 1967. MR 0217740
Bibliographic Information
- © Copyright 1982 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 274 (1982), 709-719
- MSC: Primary 53C15; Secondary 57R99, 58C05
- DOI: https://doi.org/10.1090/S0002-9947-1982-0675076-0
- MathSciNet review: 675076