The diastatic exponential of a symmetric space

Abstract

Let (M, g) be a real analytic Kähler manifold. We say that a smooth map Exp p :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 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 neighbourhood of every point of M * , the compact dual of M. We also provide a geometric interpretation of the symplectic duality map (recently introduced in Di Scala and Loi (Adv Math 217:2336-2352) 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.

Symplectic capacities of Hermitian symmetric spaces of compact and noncompact type

“…where Cut 0 (M) is the cut locus of (M, ω F S ) with respect to a fixed point 0 ∈ M (see [11,Theorem 1.1]). This diffeomorphism has been christened in [11] as a symplectic duality due to the fact that, amongst other properties, it also satisfies Φ * Ω ω 0 = ω hyp , where ω 0 denotes the standard form on C n ∼ = M \ Cut 0 (M) and ω hyp is the hyperbolic metric on Ω (see either [11] or [12] for details and also [9], [13], [29], [30], [31], [32] and [40] for the construction of explicit symplectic coordinates. The unitary ball (B 2n (1), ω 0 ) can be embedded into (Ω, ω 0 ).…”

“…Remark 13. The inclusion (31) has been obtained in [ [35] for the case of classical Cartan domains). Combining this with the symplectic embedding (29) Lu was able see [34,Theorem 1.35] to obtain the upper bound…”

Some remarks on homogeneous Kähler manifolds

“…Since the ε λ does not depend on the Kähler potential, by (9) we have ε λ (z, w) = e −λ Dz 0 (z, w) K λ Dz 0 (z, w) = C…”

Bochner Coordinates on Flag Manifolds

“…Calabi's diastasis function has shown to be the right tool in the study of the Riemannian geometry of a Kähler manifold, since it keeps track of its underlying complex and symplectic structure, in contrast with the geodesic distance which has the disadvantage to be preserved by maps between Riemannian manifolds only in the totally geodesic case, while the diastasis is preserved by any Kähler map (see [8] for a proof). For example, in [12] the first two authors of the present paper have defined the diastatic exponential, by twisting the classical riemannian one with Calabi's diastasis and obtaining applications to the symplectic geometry of symmetric spaces, while in [16,17] the second author has introduced and analyzed the concept of diastatic entropy for hyperbolic manifolds by extending that of volume entropy using the diastasis function instead of the geodesic distance. Given a point p of a Kähler manifold M, it is then natural to analyze the relationship between the maximal domain of definition of Bochner coordinates Boc p , the maximal domain of definition of the diastasis D p , say V p , and the cut locus Cut p of p (obviously Boc p ⊂ V p by the very definition of Bochner coordinates).…”