Kähler maps of Hermitian symmetric spaces into complex space forms

Abstract

In this paper, we give a complete description of the Kähler immersions of Hermitian symmetric spaces into finite or infinite dimensional complex space forms.

Kähler immersions of homogeneous Kähler manifolds into complex space forms

“…The first one due to Bochner (see Theorem 14 in [Bo47]) shows that if a Kähler manifolds can be Kähler immersed into a finite or infinite complex flat space then it can be Kähler immersed into the infinite dimensional complex projective space. The second one due to the first and the third authors of the present paper (see Lemma 3.1 in [DL07]) is a splitting result for maps into complex Euclidean spaces. We are now in the position to prove Theorem 1 which, as we have already pointed out in the introduction, will be proved assuming the validity of Theorem 4.…”

“…Assertion (2) in Theorem 1 is a generalization to arbitrary h.K.m. of Theorem 3.3 in [DL07] where the first and the third authors proved that a bounded symmetric domain which can be Kähler immersed into ℓ 2 (C) is necessarily of rank one. Actually, the method of the present paper, when applied to bounded symmetric domains, provides us with an alternative and more elegant proof of this result (cfr.…”

“…is a bounded symmetric domain. Therefore the last part of Theorem 1 provides an alternative proof of Theorem 3.3 in [DL07] without the use of Calabi's diastasis function (cfr. Remark 2).…”

The diastatic exponential of a symmetric space

“…If d p : exp p (V ) ⊂ M → R denotes the geodesic distance from p then one has: 4 and D p = d 2 p if and only if g is the flat metric. We refer the reader to the seminal paper of Calabi [3] for more details and further results on the diastasis function (see also [8,9] and [4]).…”

“…We refer the reader to the seminal paper of Calabi [3] for more details and further results on the diastasis function (see also [8,9] and [4]). …”