## A simple proof of a theorem of Calabi.(English)Zbl 1279.32009

Let $$S$$ denote one of the complex space forms $$\mathbb C^n$$, $$\mathbb P^n$$, $$\mathbb B^n$$, each equipped with its standard metric (Euclidean, Fubini-Study, Poincaré). Let $$M$$ be a connected complex manifold with real analytic Kähler metric, $$U\subset M$$ a connected open subset and $$F:U\rightarrow S$$ a holomorphic isometric embedding. The authors give a short and more or less straightforward proof (carried out for $$S=\mathbb P^n$$) that $$F$$ can be holomorphically extended along every continuous curve $$\gamma: [0,1]\rightarrow M$$ with $$\gamma(0)\in U$$; in particular $$F$$ admits a holomorphic extension $$M\rightarrow S$$ if $$M$$ is simply connected. Moreover, any isometric holomorphic embedding $$G:U\rightarrow S$$ has the form $$G=T\circ F$$ with some holomorphic isometry $$T$$ of $$S$$. The original proof for these facts, also valid for a Hilbert space form $$S$$, was given by E. Calabi [Ann. Math. (2) 58, 1-23 (1953; Zbl 0051.13103)]. For recent work in this connection see also N. Mok and S.-C. Ng [Reine Angew. Math. 669, 47–73 (2012; Zbl 1254.32033)].

### MSC:

 32D15 Continuation of analytic objects in several complex variables 53C55 Global differential geometry of Hermitian and Kählerian manifolds 32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)

### Citations:

Zbl 0051.13103; Zbl 1254.32033
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