Extrinsically Immersed Symplectic Symmetric Spaces

TL;DR

This paper characterizes extrinsic symplectic symmetric spaces via parallel second fundamental forms, proves the existence and uniqueness of full immersions, and shows how all immersions factor through these minimal ambient space embeddings.

Contribution

It establishes a complete characterization of extrinsic symplectic symmetric spaces and proves the existence, uniqueness, and minimality of their full symplectic immersions.

Findings

Extrinsic symplectic symmetric spaces have parallel second fundamental forms.

Every symmetric space with such an immersion admits a unique full immersion.

All extrinsic symplectic immersions factor through the full immersion via symplectic reduction.

Abstract

Let $(V, \Om)$ be a symplectic vector space and let $\phi: M \ra V$ be a symplectic immersion. We show that $\phi(M) \subset V$ is (locally) an extrinsic symplectic symmetric space (e.s.s.s.) in the sense of \cite{CGRS} if and only if the second fundamental form of $\phi$ is parallel. Furthermore, we show that any symmetric space which admits an immersion as an e.s.s.s. also admits a {\em full} such immersion, i.e., such that $\phi(M)$ is not contained in a proper affine subspace of $V$, and this immersion is unique up to affine equivalence. Moreover, we show that any extrinsic symplectic immersion of $M$ factors through to the full one by a symplectic reduction of the ambient space. In particular, this shows that the full immersion is characterized by having an ambient space $V$ of minimal dimension.