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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
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