Symplectic Symmetric Spaces

TL;DR

This thesis introduces and studies symplectic symmetric spaces using Lie theory and symplectic geometry, providing structural theorems, classifications, and explicit formulas for connections.

Contribution

It develops foundational theory, decomposition results, and classification theorems for symplectic symmetric spaces, extending classical Riemannian results to the symplectic setting.

Findings

Explicit formula for the Loos connection in symplectic geometry

Decomposition theorem analogous to de Rham-Wu for symplectic symmetric spaces

Complete classification of symplectic symmetric spaces in dimension four

Abstract

This is the pdf -version of the author's Ph.D. thesis (1995, ULB, Belgium). The notion of symeplectic symmertic space is introduced and studied via Lie theoretical and symplectic geoemetrical methods. The first chapter concerns basic poperties, however, an explicit formula for the Loos connection in the symplectic framework is given. In the second chapter, one proves a decomposition result analogous to de Rham - Wu's decomposition theorem for pseudo-Riemaniann symmertic spaces; this result is established for the general category of Lie triple systems as well. The chapter three deals with the reductive case; a structure theorem analoguous to Borel- de Sibenthal's theorem for Riemanniann spaces is established at the root system level. The three last chapters deal with various classes of symplectic symmetric spaces; results such as the uniqueness of a compact factor as well as a complete classification in dimension four are presented.