Rigidity of Minimal Legendrian Submanifolds in Sasakian Space Forms

This paper is concerned with the study on rigidity of minimal Legendrian submanifolds in Sasakian space forms under some certain geometric conditions, motivated by the classification of minimal Legendrian submanifolds with constant sectional curvature. First of all, we will establish a basic inequality for such submanifolds with conformally flat induced metrics and constant scalar curvature, involving the normalized scalar curvature and the squared norms of the traceless Ricci tensor and second fundamental form. Secondly, in dealing with such submanifolds with Einstein‐induced metrics, the lower bound of the squared norm of the Weyl curvature tensor is further derived using a general inequality. Specifically, these inequalities are optimal in that all Legendrian submanifolds which satisfy the equalities have been fully identified.