Totally real submanifolds and symmetric bounded domains
Let \(P_ n(c)\) be the complex projective n-space with the Kählerian metric of constant holomorphic sectional curvature \(c>0\). This paper gives a classification of n-dimensional complete totally real submanifolds M of \(P_ n(c)\) with parallel second fundamental form. Let \(D\subset C^{n+1}\) be a symmetric bounded domain of tube type realized by the Harish-Chandra imbedding. Imbed the Shilov boundary \(\hat M\) of D into the hypersphere \(S^{2n+1}(c/4)\) of the radius 2/\(\sqrt{c}\) with respect to a suitable Hermitian inner product of \({\mathbb{C}}^{n+1}\). Let \(M\subset P_ n(c)\) be the image of \(\hat M\) under the Hopf fibering \(S^{2n+1}(c/4)\to P_ n(c)\). Then M is an n-dimensional complete totally real submanifold with parallel second fundamental form and conversely such a submanifold is obtained in this way.
constant holomorphic sectional curvature
scientific article; zbMATH DE number 3872206