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MiS Preprint

Extrinsically Immersed Symplectic Symmetric Spaces

Tom Krantz and Lorenz J. Schwachhöfer


Let $(V, \Omega)$ 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 [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 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.


Related publications

2010 Repository Open Access
Tom Krantz and Lorenz J. Schwachhöfer

Extrinsically immersed symplectic symmetric spaces

In: Annals of global analysis and geometry, 37 (2010) 4, pp. 379-391