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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
59/2009

Extrinsically Immersed Symplectic Symmetric Spaces

Tom Krantz and Lorenz J. Schwachhöfer

Abstract

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.

Received:
Sep 29, 2009
Published:
Sep 30, 2009

Related publications

inJournal
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