Calabi's diastasis function for Hermitian symmetric spaces

In this paper we study the Calabi diastasis function of Hermitian symmetric spaces. This allows us to prove that if a complete Hermitian locally symmetric space (M,g)(M,g) admits a Kähler immersion into a globally symmetric space (S,G)(S,G) then it is globally symmetric and the immersion is injective. Moreover, if (S,G)(S,G) is symmetric of a specified type (Euclidean, noncompact, compact), then (M,g)(M,g) is of the same type. We also give a characterization of Hermitian globally symmetric spaces in terms of their diastasis function. Finally, we apply our analysis to study the balanced metrics, introduced by Donaldson, in the case of locally Hermitian symmetric spaces.