Isometric immersions of Kähler manifolds
A smooth map \(\varphi: M\to N\) from a Kähler manifold \(M\) to a Riemannian manifold \(N\) is said to be (1,1)-geodesic (or pluriharmonic) if the (1,1)-part of the complex bilinear extension to \(T^{\mathbb{C}} M\) of the second fundamental form \(\nabla d\varphi\) of \(\varphi\) vanishes. The authors prove that (1,1)-geodesic isometric immersions into positively 1/4-pinched Riemannian manifolds or certain conformally flat Riemannian manifolds with positive scalar curvature may only exist when \(M\) is a Riemann surface. Analogous results are obtained in the dual negative curvature cases for minimal isometric immersions. They also prove that a (1,1)-geodesic (resp. minimal) isometric immersion into the complex Grassmann manifold \(G_ p(\mathbb{C}^ n)\) (resp. into its non-compact dual \(H_ p(\mathbb{C}^ n)\)) is \(\pm\)-holomorphic if \(\dim_{\mathbb{C}} M> (p- 1)(n- p- 1)+ 1\).
scientific article; zbMATH DE number 510095
scientific article