Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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