Chen–Ricci Inequality for CR-Warped Products and Related Open Problems

Chen–Ricci inequality is derived for CR -warped products in complex space forms, Theorem 4.1 , involving an intrinsic invariant (Ricci curvature) controlled by extrinsic one (the mean curvature vector), which provides an answer for Problem 1 . As a geometric application, this inequality is applied to derive a necessary condition for the immersed submanifold to be minimal in a complex Euclidean space, which presents a partial answer for the well-known problem proposed by S.S. Chern, Problem 2 . Moreover, various applications are given. In addition, a rich geometry of CR -warped products appeared when the equality cases are discussed. Also, we extend this inequality to generalized complex space forms. In further research directions, we address a couple of open problems, namely Problems 3 and 4 .