Biharmonic Submanifolds With Parallel Normalized Mean Curvature Vector In One Class Of Product Spaces

Harmonic maps in fluid mechanics, mathematical physics equations, image pro-cessing has a wide range of applications, they are tools of researching minimal surfacein diferential geometry. Biharmonic maps are generalizations of harmonic maps.They originated in the physics of the fluid dynamics, mechanics of elasticity and thestudy of complex partial diferential equation, in terms of biology and engineeringmechanics have many important applications. Since the derivation of the biharmonicequations appeared in [19], biharmonic maps have got great development, one of thecentral topics in this area is the study of biharmonic submanifolds (i.e., those sub-manifolds whose inclusion maps are biharmonic). Many results about constructionsand classification of submanifolds in Euclidean sapces, spheres, hyperbolic spaceforms have been obtained.In this thesis, by using the methods in [6], we consider the problem of PNMC (seedef.1.1) biharmonic submanifold in the product spacesMn(c)×RorMn(c)×R1.Our main results includes the following:First, by using the equation of biharmonic submanifold in Riemannian manifoldsin [24], we deduced the equation of biharmonic equation with PNMC in productspaces.Next, we give applications of the equation in the following areas:(1) We obtained the rigidity results of PNMC biharmonic submanifolds (com-pact or complete) in Riemannian type product spaceMn(c)×R, and sufcientconditions of PNMC submanifolds become PMC submanifolds;(2) We classified PNMC biharmonic surfaces of codimension greater than onein Riemannian type product spaceMn(c)×R; (3) We obtained the rigidity results of PNMC spacelike biharmonic submanifoldsof codimension greater than one in Lorentzian product spaceMn(c)×R1;(4) We obtained sufcient conditions of spacelike PNMC biharmonic surfacesof codimension greater than one become PMC biharmonic surfaces in Lorentzianproduct spaceMn(c)×R1.