Extrinsically Immersed Symplectic Symmetric Spaces

Tom Krantz and Lorenz J. Schwachhöfer

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Submission date: 29. Sep. 2009
Pages: 17
published in: Annals of global analysis and geometry, 37 (2010) 4, p. 379-391 
DOI number (of the published article): 10.1007/s10455-009-9192-6
Bibtex
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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 [Preprint 59/2009] 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.