2009/09/29 by Tom Krantz, Krantz, Tom, Lorenz J. Schwachöfer +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG) #math.DG #math.SG
paper · pdf · doi:10.48550/arxiv.0909.5322
15 pages, version to be published by Annals of Global Analysis and Geometry
openalex publication_date 2009/09/29 · arxiv created 2009/12/18 · arxiv updated 2010/01/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (V, \Om) be a symplectic vector space and let ϕ: M \ra V be a symplectic immersion. We show that ϕ(M) ⊂ V is (locally) an extrinsic symplectic symmetric space (e.s.s.s.) in the sense of \citeCGRS if and only if the second fundamental form of ϕ 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 ϕ(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.