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Article: On holomorphic immersions into Kahler manifolds of constant holomorphic sectional curvature

TitleOn holomorphic immersions into Kahler manifolds of constant holomorphic sectional curvature
Authors
KeywordsHarmonic Form
Holomorphic Splitting
Second Fundamental Form
Tangent Sequence
Totally Geodesic Immersion
Issue Date2005
PublisherScience in China Press. The Journal's web site is located at http://www.scichina.com:8081/sciAe/EN/volumn/current.shtml
Citation
Science In China, Series A: Mathematics, 2005, v. 48 SUPPL., p. 123-145 How to Cite?
AbstractWe study holomorphic immersions f: X → M from a complex manifold X into a Kahler manifold of constant holomorphic sectional curvature M, i.e. a complex hyperbolic space form, a complex Euclidean space form, or the complex projective space equipped with the Fubini-Study metric. For X compact we show that the tangent sequence splits holomorphically if and only if f is a totally geodesic immersion. For X not necessarily compact we relate an intrinsic cohomological invariant p(x) on X, viz. the invariant defined by Gunning measuring the obstruction to the existence of holomorphic projective connections, to an extrinsic cohomological invariant v(f) measuring the obstruction to the holomorphic splitting of the tangent sequence. The two invariants p(X) and v(f) are related by a linear map on cohomology groups induced by the second fundamental form. In some cases, especially when X is a complex surface and M is of complex dimension 4, under the assumption that X 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.
Persistent Identifierhttp://hdl.handle.net/10722/156134
ISSN
2011 Impact Factor: 0.701
2012 SCImago Journal Rankings: 0.540
References

 

DC FieldValueLanguage
dc.contributor.authorMok, Nen_US
dc.date.accessioned2012-08-08T08:40:32Z-
dc.date.available2012-08-08T08:40:32Z-
dc.date.issued2005en_US
dc.identifier.citationScience In China, Series A: Mathematics, 2005, v. 48 SUPPL., p. 123-145en_US
dc.identifier.issn1006-9283en_US
dc.identifier.urihttp://hdl.handle.net/10722/156134-
dc.description.abstractWe study holomorphic immersions f: X → M from a complex manifold X into a Kahler manifold of constant holomorphic sectional curvature M, i.e. a complex hyperbolic space form, a complex Euclidean space form, or the complex projective space equipped with the Fubini-Study metric. For X compact we show that the tangent sequence splits holomorphically if and only if f is a totally geodesic immersion. For X not necessarily compact we relate an intrinsic cohomological invariant p(x) on X, viz. the invariant defined by Gunning measuring the obstruction to the existence of holomorphic projective connections, to an extrinsic cohomological invariant v(f) measuring the obstruction to the holomorphic splitting of the tangent sequence. The two invariants p(X) and v(f) are related by a linear map on cohomology groups induced by the second fundamental form. In some cases, especially when X is a complex surface and M is of complex dimension 4, under the assumption that X 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.en_US
dc.languageengen_US
dc.publisherScience in China Press. The Journal's web site is located at http://www.scichina.com:8081/sciAe/EN/volumn/current.shtmlen_US
dc.relation.ispartofScience in China, Series A: Mathematicsen_US
dc.subjectHarmonic Formen_US
dc.subjectHolomorphic Splittingen_US
dc.subjectSecond Fundamental Formen_US
dc.subjectTangent Sequenceen_US
dc.subjectTotally Geodesic Immersionen_US
dc.titleOn holomorphic immersions into Kahler manifolds of constant holomorphic sectional curvatureen_US
dc.typeArticleen_US
dc.identifier.emailMok, N:nmok@hkucc.hku.hken_US
dc.identifier.authorityMok, N=rp00763en_US
dc.description.naturelink_to_subscribed_fulltexten_US
dc.identifier.doi10.1360/05za0008en_US
dc.identifier.scopuseid_2-s2.0-20444409513en_US
dc.identifier.hkuros98234-
dc.relation.referenceshttp://www.scopus.com/mlt/select.url?eid=2-s2.0-20444409513&selection=ref&src=s&origin=recordpageen_US
dc.identifier.volume48en_US
dc.identifier.issueSUPPL.en_US
dc.identifier.spage123en_US
dc.identifier.epage145en_US
dc.publisher.placeChinaen_US
dc.identifier.scopusauthoridMok, N=7004348032en_US

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