Science in China Series A: Mathematics

, Volume 48, Supplement 1, pp 123–145 | Cite as

On holomorphic immersions into kähler manifolds of constant holomorphic sectional curvature

  • Ngaiming MokEmail author


We study holomorphic immersions f:XM from a complex manifoldX into a Kähler manifold of constant holomorphic sectional curvatureM, i.e. a complex hyperbolic space form, a complex Euclidean space form, or the complex projective space equipped with the Fubini-Study metric. ForX compact we show that the tangent sequence splits holomorphically if and only iff is a totally geodesic immersion. ForX not necessarily compact we relate an intrinsic cohomological invariantp(X) onX, viz. the invariant defined by Gunning measuring the obstruction to the existence of holomorphic projective connections, to an extrinsic cohomological invariant(f) measuring the obstruction to the holomorphic splitting of the tangent sequence. The two invariantsp(X) and?(f) are related by a linear map on cohomology groups induced by the second fundamental form. In some cases, especially whenX is a complex surface andM is of complex dimension 4, under the assumption thatX admits a holomorphic projective connection we obtain a sufficient condition for the holomorphic splitting of the tangent sequence in terms of the second fundamental form.


second fundamental form harmonic form tangent sequence totally geodesic immersion holomorphic splitting 


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Copyright information

© Science in China Press 2005

Authors and Affiliations

  1. 1.Department of MathematicsHong Kong UniversityHong Kong,China

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