Symplectic structures preserved by geodesic symmetries

Details

Supervisors

Bieliavsky, Pierre

Faculty

Faculté des sciences

Degree label

Master [120] en sciences mathématiques, à finalité approfondie

Abstract

Motivated by a theorem of Sekigawa and Vanhecke stating that Kahler manifolds with symplectic geodesic symmetries are locally symmetric, we introduce symplectic connections of type S whose geodesic symmetries preserve the symplectic form. Building on the work of D’Atri and Nickerson on Riemann manifolds with divergence-preserving symmetries (so-called D’Atri spaces) we show that connections of type S verify an infinite sequence of curvature conditions, called the odd Ledger conditions. The first of these conditions is tantamount to requiring that the Ricci curvature tensor be cyclic parallel and was used by Bourgeois and Cahen to define preferred symplectic connections, though in their case this condition was obtained through a variational principle. Furthermore, the odd Ledger conditions are sufficient in a 2-dimensional, analytic setting. This allows us to extract examples of connections of type S that are not locally symmetric from the work of Cahen and Bourgeois on preferred connections. Lastly, by confronting the Sekigawa-Vanhecke theorem to the existence of non-locally-symmetric Kähler-Einstein manifolds (which are Ricci-parallel), we obtain examples of preferred symplectic connections that are not of type S, thus proving the following sequence of strict inclusions for symplectic connections: Locally symmetric ⊂ Type S ⊂ Preferred