Extrinsically immersed symplectic symmetric spaces

Let (V, Ω) be a symplectic vector space and let \({\phi : M \rightarrow V}\) 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 Cahen et al. (J Geom Phys 59(4):409f?b-425, 2009) 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 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.