Geodesic
On Kaehler Immersions
Canadian journal of mathematics, Tome 24 (1972) no. 6, pp. 1178-1182

Voir la notice de l'article provenant de la source Cambridge University Press

Let be an (n + p)-dimensional Kaehler manifold of constant holomorphic sectional curvature . B. O'Neill [3] proved the following result.PROPOSITION A. Let M be an n-dimensional complex submanifold immersed in . If and if the holomorphic sectional curvature of M with respect to the induced Kaehler metric is constant, then M is totally geodesic.He also gave the following example: There is a Kaehler imbedding of an w-dimensional complex projective space of constant holomorphic sectional curvature 1⁄2 into an -dimensional complex projective space of constant holomorphic sectional curvature 1. This shows that Proposition A is the best possible. 
Ogiue, Koichi. On Kaehler Immersions. Canadian journal of mathematics, Tome 24 (1972) no. 6, pp. 1178-1182. doi: 10.4153/CJM-1972-126-0
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[1] 1. Ogiue, K., Differential geometry of algebraic manifolds, Differential Geometry, in honor of Yano, K., 355-372 (Kinokuniya, Tokyo, 1972). Google Scholar

[2] 2. Ogiue, K., Positively curved complex submanifolds immersed in a complex projective space (to appear in J. Differential Geometry). Google Scholar

[3] 3. O'Neill, B., Isotropic and Kaehler immersions, Can. J. Math. 17 (1965), 907–915. Google Scholar

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