On the Solvability of the Transvection group of Extrinsic Symplectic Symmetric Spaces

TL;DR

This paper proves that if a symplectic symmetric space admits a certain type of extrinsic immersion, then its transvection group must be solvable, revealing a structural property of such spaces.

Contribution

It establishes a link between extrinsic symplectic immersions and the solvability of the transvection group in symplectic symmetric spaces.

Findings

Existence of extrinsic symplectic symmetric immersion implies transvection group is solvable.

Provides conditions under which symplectic symmetric spaces have solvable transvection groups.

Abstract

Let $M$ be a symplectic symmetric space, and let $\imath : M \to V$ be an extrinsic symplectic symmetric immersion, i.e., $(V, \Omega)$ is a symplectic vector space and $\imath$ is an injective symplectic immersion such that for each point $p \in M$, the geodesic symmetry in $p$ is compatible with the reflection in the affine normal space at $\imath(p)$. We show that the existence of such an immersion implies that the transvection group of $M$ is solvable.