We show that between symmetric spaces of different types there exists a bi-symplectic map. We compute the duality map explicitely by using the theory of Jordan Algebra

Andrea Loi

Antonio J. Di Scala

Bertram

Ciriza

Cuccu

D'Ambra

Di Scala

Gromov

Jacobson

Kobayashi

Koecher

McCrimmon

McDuff

Roos

Satake

Takeuchi

Wolf

English

AbstractLet M⊂Cn be a complex n-dimensional Hermitian symmetric space endowed with the hyperbolic form ωhyp. Denote by (M∗,ωFS) the compact dual of (M,ωhyp), where ωFS is the Fubini–Study form on M∗. Our first result is Theorem 1.1 where, with the aid of the theory of Jordan triple systems, we construct an explicit symplectic duality, namely a diffeomorphism ΨM:M→R2n=Cn⊂M∗ satisfying ΨM∗ω0=ωhyp and ΨM∗ωFS=ω0 for the pull-back of ΨM, where ω0 is the restriction to M of the flat Kähler form of the Hermitian positive Jordan triple system associated to M. Amongst other properties of the map ΨM, we also show that it takes (complete) complex and totally geodesic submanifolds of M through the origin to complex linear subspaces of Cn. As a byproduct of the proof of Theorem 1.1 we get an interesting characterization (Theorem 5.3) of the Bergman form of a Hermitian symmetric space in terms of its restriction to classical complex and totally geodesic submanifolds passing through the origin

Di Scala, Antonio J.

Loi, Andrea

Elsevier - Publisher Connector 

Symplectic duality of symmetric spaces 

Crossref

Symplectic duality of symmetric spaces

Di Scala, Antonio Jose'

PORTO Publications Open Repository TOrino

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Symplectic duality of Symmetric Spaces

DI SCALA, ANTONIO JOSE'

ANDREA LOI

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

Let M  \subset {\complex}^n be
a complex n-dimensional Hermitian symmetric space endowed  with
the hyperbolic form \omega_{hyp}. Denote by  (M^*, \omega_{FS}) the compact dual of  (M, \omega_{hyp}), where\omega_{FS} is the Fubini--Study form on M^*. Our first result
is Theorem 1 where, with the aid of the theory of Jordan triple systems, we construct an explicit {\em symplectic
duality}, namely a diffeomorphism \Psi_M: M\rightarrow
{\real}^{2n}={\complex}^n\subset M^* satisfying
\Psi_M^*\omega_0=\omega_{hyp} and
\Psi_M^*\omega_{FS}=\omega_0  for the pull-back of \Psi_M, where \omega_0
is the restriction to M of the flat Kaehler  form of
the Hermitian
positive Jordan triple system  associated to M.
Amongst other properties of the
map \Psi_M, we also show that it takes (complete) complex and
totally geodesic submanifolds of $M$ through the origin to complex
linear subspaces of {\complex}^n. As a byproduct of the proof of
Theorem \ref{mainteor} we get an interesting characterization
of the Bergman form of a Hermitian
symmetric  space in terms of its restriction to classical complex
and totally geodesic submanifolds passing through the origin

LOI, ANDREA

DI SCALA A. J.

Archivio istituzionale della ricerca - Università di Cagliari

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Symplectic duality of Symmetric Spaces

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Elsevier - Publisher Connector

Crossref

PORTO Publications Open Repository TOrino

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

Archivio istituzionale della ricerca - Università di Cagliari

Elsevier - Publisher Connector