Symplectic duality of symmetric spaces
LOI, ANDREA;
2008-01-01
Abstract
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.Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.