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Proceedings of the American Mathematical Society

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Projectively flat affine surfaces that are not locally symmetric

Author: Isaac Chaujun Lee
Journal: Proc. Amer. Math. Soc. 123 (1995), 237-246
MSC: Primary 53A15; Secondary 53C05, 53C40
MathSciNet review: 1212285
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Abstract: By studying affine rotation surfaces (ARS), we prove that any surface affine congruent to $ {x^2} + \epsilon {y^2} = {z^r}$ or $ {y^2} = z(x + \epsilon z\log z)$ is projectively flat but is neither locally symmetric nor an affine sphere, where $ \epsilon$ is 1 or $ - 1, r \in {\mathbf{R}} - \{ - 1,0,1,2\} $, and $ z > 0$. The significance of these surfaces is due to the fact that until now $ {x^2} + \epsilon {y^2} = {z^{ - 1}}$ are the only known surfaces which are projectively flat but not locally symmetric. Although Podestà recently proved the existence of an affine surface satisfying the above italicized conditions, he did not construct any concrete example.

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  • [KN] S. Kobayashi and K. Nomizu, Foundations of differential geometry, Vol. I, Wiley, New York, 1969.
  • [L1] C. Lee, Affine rotation surfaces, Master's Thesis, Brown Univ., 1991.
  • [L2] -, Generalized affine rotation surfaces, Ph.D. Thesis, Brown Univ., 1993.
  • [MR] M. Magid and P. Ryan, Flat affine spheres in $ {{\mathbf{R}}^3}$, Geom. Dedicata 33 (1990), 277-288. MR 1050415 (91e:53016)
  • [N1] K. Nomizu, What is affine differential geometry, Proc. Conference on Diff. Geom., Munster, 1982, pp. 42-43.
  • [N2] -, Introduction to affine differential geometry, Part I, Lecture Notes, MPI preprint MPI 88-37, 1988; revised: Department of Mathematics, Brown University, 1989.
  • [NP] K. Nomizu and U. Pinkall, On a certain class of homogeneous flat manifolds, Tôhoku Math. J. 39 (1987), 407-427. MR 902579 (88j:53050)
  • [Po] F. Podestà, Projectively flat surfaces in $ {\mathbb{A}^3}$, Proc. Amer. Math. Soc. 119 (1993), 255-260. MR 1169045 (93k:53022)
  • [S] U. Simon, Local classification of two-dimensional affine spheres with constant curvature metric, Differential Geometry Appl. 1 (1991), 123-132. MR 1244439 (94g:53006)

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Keywords: Blaschke immersion, locally symmetric, projectively flat, affine rotation surfaces
Article copyright: © Copyright 1995 American Mathematical Society

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