CR-Submanifolds in Complex and Sasakian Space Forms

Abstract

We survey recent results on CR-submanifolds in complex space forms and contact CR-submanifolds in Sasakian space forms, including a few contributions of the present authors. The Ricci curvature and k-Ricci curvature of such submanifolds are estimated in terms of the squared mean curvature. A Wintgen-type inequality for totally real surfaces in complex space forms is proved. The equality case is shown to hold if and only if the ellipse of curvature is a circle at every point of the surface; an example of a totally real surface in \(\mathbf{C}^2\) satisfying the equality case identically is provided. A generalized Wintgen inequality for Lagrangian submanifolds in complex space forms is established. A Wintgen-type inequality for CR-submanifolds in complex space forms is stated. Geometric inequalities for warped product submanifolds in complex space forms are proved, the equality cases are characterized and examples for the equality cases are given. Also some obstructions to the minimality of warped product CR-submanifolds in complex space forms are derived. The scalar curvature of such submanifolds is estimated and classifications of submanifolds in complex space forms satisfying the equality case are given. The survey ends with results on contact CR-submanifolds in Sasakian space forms. Geometric inequalities for the Ricci curvature and k-Ricci curvature of contact CR-submanifolds in Sasakian space forms are stated. A recent result by the second author (i.e., a generalized Wintgen inequality for C-totally real submanifolds in Sasakian space forms) is extended to contact CR-submanifolds in Sasakian space forms.