Characterizations of real hypersurfaces in complex space forms in terms of curvature tensors
The authors provide characterizations of certain real hypersurfaces in a complex projective space \(P_n \mathbb{C}\) or a complex hyperbolic space \(H_n\mathbb{C}\) in terms of Riemannian curvature. Let \(M\) be such a hypersurface with induced Riemannian metric \(g\), Riemannian curvature tensor \(R\) and Riemannian connection \(\nabla\). Moreover, let \(\xi\) be the vector field on \(M\) obtained by applying the Kähler structure of the ambient space to a unit normal field of \(M\), and \(A\) the shape operator of \(M\) with respect to \(\xi\). The authors obtain the following results: (1) If (a) \(n\geq 3\), \(\nabla_\xi R= 0\) and \(g(A\xi, \xi)\neq 0\), or if (b) \(n\geq 2\) and the Lie derivative of \(R\) in direction of \(\xi\) vanishes, then \(M\) is an open part of a tube around a totally geodesic complex subspace of \(P_n \mathbb{C}\) or \(H_n\mathbb{C}\), or of a horosphere in \(H_n \mathbb{C}\). (2) If \(n\geq 3\), \(\nabla_\xi R= 0\), and if the ambient space is \(P_n \mathbb{C}\), then \(M\) is a non-homogeneous real hypersurface lying on a tube of a certain radius around a certain Kähler submanifold in \(P_n\mathbb{C}\), or is an open part of a tube around a totally geodesic complex subspace of \(P_n \mathbb{C}\).
scientific article; zbMATH DE number 828344