A possible symplectic framework for Radon-type transforms

TL;DR

This paper proposes a symplectic geometric framework for Radon-type transforms, utilizing symplectic symmetric spaces with Ricci-type connections to generalize classical integral transforms.

Contribution

It introduces a novel symplectic setting for Radon transforms based on symmetric spaces with Ricci-type connections, extending classical concepts to symplectic geometry.

Findings

Defined Radon-type transforms in symplectic symmetric spaces.

Identified dual pairs of spaces involving totally geodesic symplectic submanifolds.

Established a framework for symplectic Radon transforms and their duality.

Abstract

Our project is to define Radon-type transforms in symplectic geometry. The chosen framework consists of symplectic symmetric spaces whose canonical connection is of Ricci-type. They can be considered as symplectic analogues of the spaces of constant holomorphic curvature in K\"ahlerian Geometry. They are characterized amongst a class of symplectic manifolds by the existence of many totally geodesic symplectic submanifolds. We present a particular class of Radon type tranforms, associating to a smooth compactly supported function on a homogeneous manifold $M$, a function on a homogeneous space $N$ of totally geodesic submanifolds of $M$, and vice versa. We describe some spaces $M$ and $N$ in such Radon-type duality with $M$ a model of symplectic symmetric space with Ricci-type canonical connection and $N$ an orbit of totally geodesic symplectic submanifolds.