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Kähler submanifolds whose Ricci curvature is bounded from below. (English) Zbl 1098.32013

An \(n\)-dimensional complex projective space \(n\)-\(\mathbb{C} P(c)\) is an \(n\)-dimensional complex Kähler manifold of constant holomorphic sectional curvature \(c > 0\). K. Ogiue [Hokkaido Math. J. 1, 16–20 (1972; Zbl 0246.53055)] shows a sufficient condition (the Ricci curvatures are greater than \(n/2)\) for an \(n\)-dimensional complete Kähler submanifold, of an \(n+p\)-\(\mathbb{C} P(1)\), to be totally geodesic. H. Nakagawa and R. Takagi [J. Math. Soc. Japan 28, 638–667 (1976; Zbl 0328.53009)] classified the compact Kähler \(n\)-submanifolds immersed in an \(m\)-\(\mathbb{C} P(1)\) with parallel second fundamental form.
The authors give the classification of complete Kähler submanifolds immersed in \(n+p\)-\(\mathbb{C} P(1)\), whose Ricci curvature is greater than or equal to \(n/2\). The first proof consists in the evaluation of the Laplacian of the squared norm of the second fundamental tensor. Another proof is obtained by using Ros’s theorem for the holomorphic pinching condition \(H\) (the holomorphic sectional curvature) greater than or equal to \(1/2\).

MSC:

32Q40 Embedding theorems for complex manifolds
32Q20 Kähler-Einstein manifolds
32Q57 Classification theorems for complex manifolds
53C40 Global submanifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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References:

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