Extrinsically immersed symplectic symmetric spaces

Let ( V , Ω) be a symplectic vector space and let be a symplectic immersion. We show that is locally an extrinsic symplectic symmetric space (e.s.s.s.) in the sense of Cahen et al. (J Geom Phys 59(4):409f́b-425, 2009) if and only if the second fundamental form of 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 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.