Extrinsically Immersed Symplectic Symmetric Spaces

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.