Extrinsically Immersed Symplectic Symmetric Spaces

Details Extrinsically Immersed Symplectic Symmetric Spaces Extrinsically Immersed Symplectic Symmetric Spaces

2009-09-29

arxiv.org/abs/0909.5322v2

Mathematics

Symplectic Geometry

15 pages, version to be published by Annals of Global Analysis and Geometry

Scientific paper

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.

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