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We give criterions for a submanifold to be an extrinsic sphere and to be a totally geodesic submanifold by observing some Frenet curves of order 2 on the submanifold.
We also characterize constant isotropic immersions into arbitrary Riemannian manifolds in terms of Frenet curves of proper order 2 on submanifolds.
As an application we obtain a characterization of Veronese embeddings of complex projective spaces into complex projective spaces.
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참고문헌 (8)
참고문헌 신청
B. O'Neill / 1965 / Isotropic and Kahler immersions / Canad. J. Math. 17 : 907 ~ 915
E. Calabi / 1953 / Isometric imbedding of complex manifolds / Ann. of Math. 58 : 1 ~ 23
H. Nakagawa / 1976 / Complex space forms immersed in complex space forms / Trans. Amer. Math. Soc. 219 : 289 ~ 297
H. Tanabe / 2006 / Characterization of totally geodesic submanifolds in terms of Frenet curves / Sci. Math. Jpn. 63 (1) : 83 ~ 88
K. Nomizu / 1974 / On circles and spheres in Riemannian geometry / Math. Ann. 210 : 163 ~ 170
E. Calabi / 1953 / Isometric imbedding of complex manifolds / Ann. of Math. 58 : 1 ~ 23
H. Nakagawa / 1976 / Complex space forms immersed in complex space forms / Trans. Amer. Math. Soc. 219 : 289 ~ 297
H. Tanabe / 2006 / Characterization of totally geodesic submanifolds in terms of Frenet curves / Sci. Math. Jpn. 63 (1) : 83 ~ 88
K. Nomizu / 1974 / On circles and spheres in Riemannian geometry / Math. Ann. 210 : 163 ~ 170
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