Holomorphic retractions of bounded symmetric domains onto totally geodesic complex submanifolds
This paper is a contribution to the geometry of bounded symmetric domains. The main result is as follows: Let \(\Omega \subset \mathbb{C}^n\) be such a domain, and \(G\) the identity component of the holomorphic automorphisms of \(\Omega \). Let \(E\subset \mathbb{C}^n\) be a complex vector subspace such that \(S=E\cap \Omega\) is a totally geodesic complex submanifold with respect to a \(G\)-invariant Kähler metric. Then \(\rho (\Omega )=S\), where \(\rho \) is the orthogonal projection \(\mathbb{C}^n\to E\). A special case has been considered previously by Yeung. In that case \(\Omega \) is the irreducible bounded symmetric domain in the space of symmetric complex matrices and \(S\) is isomorphic to the unit ball in the space \(M(a,b;\mathbb{C})\) of \(a\times b\) complex matrices. Furthermore the totally geodesic complex submanifolds in \(\Omega \) are characterized in terms of Jordan triple systems. The author gives also a characterization of the Harish-Chandra realization of bounded symmetric domains in terms of bisectional curvatures.
scientific article; zbMATH DE number 7629821
scientific article