Extrinsically Immersed Symplectic Symmetric Spaces
Tom Krantz, Lorenz J. Schwach\"ofer

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 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 \cite{CGRS} 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 {\em full} such immersion, i.e., such that is not contained in a proper affine subspace of , and this immersion is unique up to affine equivalence. Moreover, we show that any extrinsic symplectic immersion of 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 of minimal dimension.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic Geometry and Number Theory · Advanced Algebra and Geometry
