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Pointed Blaschke manifolds and geodesic normal sections

Published online by Cambridge University Press:  17 April 2009

Dong-Soo Kim ,
Young Ho Kim  and
Eun Kyoung Lee
Dong-Soo Kim
Affiliation:
Department of Mathematics, Connam National University, K wangju 535–737, Korea e-mail: [email protected]
Young Ho Kim
Affiliation:
Department of Mathematics, Kyungpook National University, Taegu 702–701, Korea e-mail: [email protected]
Eun Kyoung Lee
Affiliation:
Department of Mathematics, Connam National University, K wangju 535–737, Korea e-mail: [email protected]
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We study complete submanifolds of Euclidean space where every geodesic passing through a fixed point is the normal section along it. We prove that all such geodesics are independent of the direction at the point and such submanifolds are pointed Blaschke manifolds or diffeomorphic to a Euclidean space.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 70 , Issue 3 , December 2004 , pp. 377 - 383
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Besse, A., Manifolds all of whose geodesics are closed (Springer-Verlag, Berlin, Heidelberg New York, 1978).Google Scholar
[2]Chen, B.-Y., Geometry of submanifolds (Marcel Dekker, New York, 1973).Google Scholar
[3]Chen, B.-Y. and Verheyen, P., ‘Submanifolds with geodesic normal sections’, Math. Ann. 269 (1984), 417–429.CrossRefGoogle Scholar
[4]Fueki, S., ‘Pointed helical submanifolds of pseudo-Riemannian manifolds’, J. Geom. 62 (1998), 129–143.CrossRefGoogle Scholar
[5]Hong, S.L., ‘Isometric immersions of manifolds with plane geodesics into Euclidean space’, J. Differential Geom. 8 (1973), 259–278.Google Scholar
[6]Kim, Y.H., ‘Surfaces of Euclidean spaces with planar or helical geodesics through a point’ Ann. Mat. Pura. Appl. 164 (1993), 1–35.Google Scholar
[7]Kim, Y.H., ‘Pointed helical submanifolds of Euclidean space’, J. Geom. 50 (1994). 111–117.CrossRefGoogle Scholar
[8]Kim, Y.H., ‘Pointed planar geodesic submanifolds in Euclidean space’, Vietnam J. Math. 31 (2003), 1–8.Google Scholar
[9]Little, A., ‘Manifolds with planar geodesics’, J. Differential Geom. 11 (1976), 265–285.CrossRefGoogle Scholar
[10]O'Neill, B., ‘Isotropic and Kaehler immersions’, Canad. J. Math. 17 (1965), 907–915.Google Scholar
[11]Sakamoto, K., ‘Planar geodesic immersions’, Tôhoku Math. J. 29 (1977), 25–56.Google Scholar
[12]Sakamoto, K., ‘Helical immersions into a Euclidean space’, Michigan Math. J. 33 (1986), 353–364.CrossRefGoogle Scholar
[13]Sakamoto, K., ‘Helical immersions into a unit sphere’, Math. Ann. 261 (1982), 63–80.CrossRefGoogle Scholar
[14]Tsukada, K., ‘Helical geodesic immersions of compact rank one symmetric spaces into spheres’, Tokyo J. Math. 6 (1983), 267–285.CrossRefGoogle Scholar
[15]Verheyen, P., ‘Submanifolds with geodesic normal sections are helical’, Rend. Sem. Mat., Univ. Politec. Torino 43 (1985), 511–527.Google Scholar