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Axioms for Elliptic Geometry

  • David Gans (a1)

Extract

Until recently the literature contained little on the axiomatic foundations of elliptic geometry that was non-analytical and independent of projective geometry. During the past decade this subject has come in for further study, notably by Busemann [2] and Blumenthal [1], who supplied such foundations. This paper presents another and, it is believed, simpler effort in the same general direction, proceeding by the familiar synthetic methods of elementary geometry and using only elementary topological notions and ideas concerning metric spaces. Specifically, elliptic 2-space is obtained on the basis of six axioms, most notable of which is one assuming the existence of translations. The writer wishes to express his deep appreciation to Herbert Busemann for his invaluable help.

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Copyright

References

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1. Blumenthal, L. M., Metric characterization of elliptic space, Trans. Amer. Math. Soc, vol. 59 (1946), 381400.
2. Busemann, Herbert, Metric methods in Finsler spaces and in the foundations of geometry (Ann. Math. Studies, No. 8, Princeton, 1942).
3. Fenchel, W., Elementare Beweise und Anwendungen einiger Fixpunktsatze, Meit. Tids., B (1932), 6687.
4. Menger, K., Untersuchungen über allgemeine Metrik, Math. Ann., vol. 100 (1928), 74113.
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