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A surface which contains planar geodesics

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Abstract

It is proved that a complete surface in E 3 is a sphere or a plane if it contains at least four geodesics through each point which are plane curves.

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References

  1. Spivak, M., A Comprehensive Introduction to Differential Geometry, III, Publish or Perish, 1975.

  2. Ogiue, K. and Takagi, R., ‘A Submanifold which contains Many Extrinsic Circles’, Tsukuba J. Math. 8 (1984), 171–182.

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  3. Takeuchi, N., ‘A Sphere as a Surface which contains Many Circles’, J. Geom. 24 (1985), 123–130.

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Authors and Affiliations

  1. Department of Mathematics, Tokyo Metropolitan University, Setagaya, 158, Tokyo, Japan

    Nobuko Takeuchi

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  1. Nobuko Takeuchi
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Takeuchi, N. A surface which contains planar geodesics. Geom Dedicata 27, 223–225 (1988). https://doi.org/10.1007/BF00151355

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  • DOI: https://doi.org/10.1007/BF00151355

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