The diastatic exponential of a symmetric space

TL;DR

This paper introduces the concept of diastatic exponential maps on Hermitian symmetric spaces, proving their existence, uniqueness, and global properties, and relates them to symplectic duality and reproducing kernels.

Contribution

It establishes the existence and uniqueness of globally defined diastatic exponential maps on Hermitian symmetric spaces of noncompact type and their duals, linking them to symplectic duality and reproducing kernels.

Findings

Existence of globally defined diastatic exponential maps on noncompact Hermitian symmetric spaces.

Uniqueness of these maps determined by their restriction to polydisks.

Connection between diastatic exponentials and the symplectic duality map.

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 $T_pM$ 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 $d^2_p$ 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 $M^*$, 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 $M^*$ to the reproducing kernel of $M$.