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Kähler maps of Hermitian symmetric spaces into complex space forms. (English) Zbl 1132.53040
The authors study Kähler immersions of Hermitian symmetric spaces into finite- or infinite-dimensional complex space forms, in particular Kähler immersions of such spaces of noncompact type endowed with their Bergman metrics into the infinite-dimensional hyperbolic space $$\mathbb{C} H^\infty$$ or the infinite-dimensional Euclidean space $$\ell^2(\mathbb{C})$$. Inspired by Calabi, they show: among all Hermitian symmetric spaces of noncompact type, the products of hyperbolic spaces are the only ones which admit Kähler immersions into $$\mathbb{C} H^\infty$$ or $$\ell^2(\mathbb{C})$$. This, together with known results, leads to a description of those Hermitian symmetric spaces which admit a Kähler immersion into a given complex space form.

##### MSC:
 53C55 Global differential geometry of Hermitian and Kählerian manifolds 58C25 Differentiable maps on manifolds 53C40 Global submanifolds
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