Classification And Rigidity Of Legendrian Submanifolds In The Unit Sphere

Geometry of submanifolds is one of the most important research directions in differential geometry.Contact manifold is a class of important research objects,and the study on its Legendrian submanifolds has attracted much attention.In this pa-per,we study the classification problem of the Legendrian submanifolds in Sasakian space form S7(1)and the rigidity problem for Legendrian surfaces in Sasakian space form S5(1),respectively.Our main results are as follows:(1)We study the minimal Willmore Legendrian submanifold in Sasakian space form S7(1)with the constant scalar curvature.With the assumption of constant scalar curvature,by employing the minimal Legendrian condition which get the important lemma about Willmore submanifolds(see Lemma 3.1.7).Moreover,we give some examples of minimal Willmore Legendrian submanifolds of Sasakian space form S7(1)with constant scalar curvature.Thus,we completely classify such kind of submanifolds(see Theorem 1.3).(2)We derive the rigidity theorem for compact Legendrian H-surfaces in Sasakian space form S5(1).We establish an inequality of Simons' type for Legendri-an surfaces of S5(1).As the more in-depth,we get the inequality relation between the integral of a function of p2 and the Euler characteristic number of ?(M)in which the equality holds if and only if the Legendrian H-surfaces is either a totally geodesic sphere S2(1)or a minimal flat Legendrian torus T?(see Theorem 1.6).