Pinching Problems Of Minimal Submanifolds In Product Manifolds

In the theory of submanifolds,the interplay between geometry and topology has always been the fundamental research topic and plays an important role in understanding both fields.In this thesis,we mainly investigate the geometric rigidity of minimal submanifolds in product manifolds.This paper consists of three parts and is organized as follows.In chapter 2,we prove a series of pinching theorems for compact minimal submanifolds in a generalized cylinder Sn1(c)× R2.Actually,by using the Simons' type equations for various geometrical invariants,we obtain several pinching theorems concerning the squared length of the second fundamental form,the Ricci curvature and the squared maximum norm of the second fundamental form.In chapter 3,we prove a new DDVV type inequality for immersed submanifolds into Mn1(c)× Rn2.Furthermore,we obtain a pinching theorem in Sn1(c)× Rn2,which extends Lu's and Chen-Cui's pinching theorems.In chapter 4,we study the rigidity of compact minimal submanifolds in Riemannian warped products I×f Sm(c).We derive the compatible equations for the immersed submanifolds of I×f Mm(c),and obtain a lower bound of the squared length of the first covariant derivative of the second fundamental form.Then we prove a new pinching theorem for compact minimal submanifolds immersed in I ×f Sm(c).This result allows us to generalize Chen-Cui's pinching theorem from Riemannian products Sm(c)× R to Riemannian warped products I×f Sm(c).