Let M1 and M2 be two K¨ahler manifolds. We call M1 and M2 relatives if they share a non-trivial K¨ahler submanifold S, namely, if there exist two holomorphic and isometric immersions (K¨ahler immersions) h1 : S → M1 and h2 : S → M2. Moreover, two K¨ahler manifolds M1 and M2 are said to be weakly relatives if there exist two locally isometric (not necessarily holomorphic) K¨ahler manifolds S1 and S2 which admit two K¨ahler immersions into M1 and M2 respectively. The notions introduced are not equivalent (cf. Example 2.3). Our main results in this paper are Theorem 1.2 and Theorem 1.4. In the first theorem we show that a complex bounded domain D ⊂ Cn with its Bergman metric and a projective K¨ahler manifold (i.e. a projective manifold endowed with the restriction of the Fubini–Study metric) are not relatives. In the second theorem we prove that a Hermitian symmetric space of noncompact type and a projective K¨ahler manifold are not weakly relatives. Notice that the proof of the second result does not follows trivially from the first one. We also remark that the above results are of local nature, i.e. no assumptions are used about the compactness or completeness of the manifolds involved.
Kahler manifolds and their relatives / Di Scala A.J.; Loi A.. - In: ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA. CLASSE DI SCIENZE. - ISSN 0391-173X. - IX:5(2010), pp. 495-501.
Titolo: | Kahler manifolds and their relatives | |
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Data di pubblicazione: | 2010 | |
Rivista: | ||
Appare nelle tipologie: | 1.1 Articolo in rivista |
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N35Porto.pdf | Here is the email from the editor allowing the upload of the preprint: Gent.mo Prof. Di Scala, mi rincresce doverla informare che l’accesso gratuito agli articoli della nostra rivista è previsto solo per i volumi pubblicati con un lasso di tempo di 5 anni (in sostanza al momento si possono scaricare gratuitamente lavori pubblicati fino al volume 5/2006). Tuttavia può inserire il preprint indicando che si tratta di un articolo pubblicato negli Annali SNS. Cordiali saluti, Luisa Ferrini | 1. Preprint / submitted version [pre- review] | PUBBLICO - Tutti i diritti riservati | Visibile a tuttiVisualizza/Apri |
http://hdl.handle.net/11583/2264095