Symplectic structures preserved by geodesic symmetries

TL;DR

This paper extends previous results by showing that symplectic manifolds with geodesic symmetries preserving the symplectic form are a special class called 'S-type' affine symplectic manifolds, relaxing the hermitian condition.

Contribution

It generalizes the concept of symmetry preservation from hermitian to purely symplectic manifolds, defining and studying 'S-type' affine symplectic manifolds.

Findings

Identification of 'S-type' symplectic manifolds with preserved geodesic symmetries.

Extension of symmetry preservation results to non-hermitian symplectic manifolds.

Characterization of symplectic manifolds with symplectic connection and local symplectomorphism symmetries.

Abstract

Answering a conjecture by S. Kobayashi, in 1986, K. Sekigawa and L. Vanhecke proved that an almost hermitian manifold whose local geodesic symmetries preserve the K\"ahler 2-form is a locally symmetric hermitian space. In the present paper, we relax the hermitean hypothesis by only requiring the manifold to be symplectic. In other words, we study the symplectic manifolds equipped with a symplectic connection whose geodesic symmetries are (local) symplectomorphisms. We call ``S-type'' these affine symplectic manifolds.