The diastatic exponential of a symmetric space

Abstract

Let $(M,g)$ be a real analytic Kaehler manifold. We say that a smooth map $E_p:W\to M$ from a neighborhood $W$ of the origin of $T_{p}M$ into $M$ is a {\em diastatic exponential} at $p$ if it satisfies $$(d \E_p)_0=\id_{T_pM},$$ $$D_p(\E_p (v))=g_p(v, v), \forall v\in W,$$ where $D_p$ is Calabi's diastasis function at $p$ (the usual exponential ${\exp }_p$ obviously satisfied these equations when $D_p$ is replaced by the square of the geodesics distance $d_p^2$ from $p$). In this paper we prove that for every point $p$ of an Hermitian symmetric space of noncompact type M there exists a globally defined diastatic exponential centered in $p$ which is a diffeomorphism and it is uniquely determined by its restriction to polydisks. An analogous result holds true in an open dense neighborhood of every point of $M^{\ast }$, the compact dual of $M$. We also provide a geometric interpretation of the symplectic duality map in terms of diastatic exponentials. As a byproduct of our analysis we show that the symplectic duality map pulls back the reproducing kernel of $M^{\ast }$ to the reproducing kernel of $M$.

11 pages

Cited by in corpus (9)

• Some remarks on homogeneous Kähler manifolds
• The log-term of the disc bundle over a homogeneous Hodge manifold
• The volume entropy of local Hermitian symmetric space of noncompact type
• A note on diastatic entropy and balanced metrics
• Bochner coordinates on flag manifolds
• Symplectic geometry of Cartan-Hartogs domains
• Minimal symplectic atlases of Hermitian symmetric spaces
• A Cartan-Hartogs version of the Polydisk Theorem
• On the $\Delta$-property for complex space forms