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Calabi’s diastasis function for Hermitian symmetric spaces. (English) Zbl 1096.53043
On a complex manifold $$M$$, with a real analytic Kähler metric $$g$$, on a neighborhood of every point $$p\in M$$, one can define the diastasis function [E. Calabi, Ann. Math. (2) 58, 1–23 (1953; Zbl 0051.13103)]. In this paper, the author studies the diastasis function of Hermitian symmetric spaces. The main results are generalisations of results of M. Takeuki [Jap. J. Math. 4, 171–219 (1978; Zbl 0389.53027)] and H. Nakagawa and R. Takagi [J. Math. Soc. Japan 28, 638–667 (1976; Zbl 0328.53009)], that is “if there exists a Kähler immersion of a complete Hermitian locally symmetric space $$(M,g)$$ into an almost projective like Kähler manifold $$(S,G)$$ then $$M$$ is forced to be simply connected and the immersion to be injective. Moreover if $$(S,G)$$ is globally symmetric and of a given type (Euclidean, noncompact, compact) then $$(S,G)$$ is of the same type.” Also, a characterisation of Hermitian globally symmetric spaces in terms of their diastasis function is given.

##### MSC:
 53C55 Global differential geometry of Hermitian and Kählerian manifolds 53C35 Differential geometry of symmetric spaces
##### Keywords:
Kähler metrics; diastasis function; symmetric space
Full Text:
##### References:
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