The idea of considering the second fundamental form of a hypersurface as the first fundamental form of another hypersurface has found very useful applications in Riemannian and semi-Riemannian geometry, especially when trying to characterize extrinsic hyperspheres and ovaloids. Recently, T. Adachi and S. Maeda gave a characterization of totally umbilical hypersurfaces in a space form by circles. In our paper, we give a characterization of totally umbilical hypersurfaces of a space form by means of geodesic mapping.
References
S. Verpoort, The Geometry of the Second Fundamental Form: Curvature Properties and Variational Aspects, Ph. D Thesis, Katholieke Univ. Leuven (2008).
T. Adachi and S. Maeda, “Characterization of totally umbilic hypersurfaces in a space form by circles,” Czechoslovak Math. J., 55, No. 1, 203–207 (2005).
J. Gerretsen, Lectures on Tensor Calculus and Differential Geometry, Noordhoff, Groningen (1962).
C. E. Weatherburn, An Introduction to Riemannian Geometry and the Tensor Calculus, Cambridge Univ. Press, Cambridge (1966).
K. Yano and M. Kon, Structures on Manifolds, World Scientific, Singapore (1984).
B. Y. Chen, Geometry of Submanifolds, Marcel Dekker, New York (1973).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 4, pp. 583–587, April, 2013.
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Canfes, E.Ö., Özdeğer, A. A characterization of totally umbilical hypersurfaces of a space form by geodesic mapping. Ukr Math J 65, 643–648 (2013). https://doi.org/10.1007/s11253-013-0801-y
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DOI: https://doi.org/10.1007/s11253-013-0801-y