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A characterization of the Veronese varieties*

Published online by Cambridge University Press:  22 January 2016

Katsumi Nomizu
Affiliation:
Brown University
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Extract

Let Pm(C) be the complex projective space of dimension m. In a previous paper [2] we have proved

THEOREM A. Let f be a Kaehlerian immersion of a connected, complete Kaehler manifold Mn of dimension n into Pm(C). If the image f(τ) of each geodesic τ in Mn lies in a complex projective line P1(C) of Pm(C), then f(Mn) is a complex projective subspace of Pm(C), and f is totally geodesic.

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1976

Footnotes

*

Work supported by NSF Grant GP-38582X.

References

[1] Cecil, T. E. : Geometric applications of critical point theory to submanifolds of complex projective space, Nagoya Math. J., 55 (1974), 531.CrossRefGoogle Scholar
[2] Nomizu, K. : Eine Kennzeichnung der projektiven Unterräume in dem komplexen projektiven Raum nach ihren Geodätischen, Math. Z., 137 (1974), 147150.CrossRefGoogle Scholar
[3] Nomizu, K. and Smyth, B. : Differential geometry of complex hypersurfaces II, J. Math. Soc. Japan, 20 (1968), 498521.CrossRefGoogle Scholar
[4] Nomizu, K. and Yano, K. : On circles and spheres in Riemannian geometry, Math. Ann., 210 (1974), 163170.CrossRefGoogle Scholar
[5] Ogiue, K. : Differential Geometry of Kaehler Submanifolds, Advances in Mathematics, vol. 13, no. 1 (1974), 73114.CrossRefGoogle Scholar
[6] O’Neill, B.: Isotropic and Kähler immersions, Canadian J. Math., 17(1965), 907915.Google Scholar

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