The Infinitesimal Rigidity Of Hypersurfaces In Space Forms

The isometric embedding problem is one of the fundamental problems in differential geometry.Since Riemannian manifold was formulated by Riemman in 1868,naturally there arose the question whether an abstract Riemmannian manifold is simply a submanifold of some Euclidean space with its induced metric.In other words,it's the question of reality of Riemannian manifold.As known the uniqueness of solution in partial differential equations is related to the existence,hence it's another important topic.The counterpart of uniqueness in isometric embedding is global rigidity.The linearized version of global rigidity is infinitesimal rigidity.It plays great roles in the linearized problem in the isometric embedding problem.In the present paper,we revisit the infinitesimal rigidity of hypersurfaces in space forms.In the first part of the paper,We highlight Darboux equations satisfied by the function ?=1/2(?) and its linearized version ?= (?) where (?) is the isometric deformation up to first order and give new proofs of the infinitesimal rigidity of hypersurfaces by energy method and maximal principle.In the new proofs,we see that ? and ? carry all the information of a surface with prescribed metric.At the same time,we observe the resemblance of the rigidity in prescribed curvature problem R3 to the infinitesimal rigidity in isometric embedding problem,so we borrow the idea of Guang-Wang-Zhang[1]which proved the rigidity in prescribed curvature problem to turn the infinitesimal rigidity into the uniqueness of solution to an elliptic partial differential equation and obtain the uniqueness by maximal principle.In the paper,we apply the method to non closed case:Alexandrov's positive annuli.In the present paper,we also prove the equivalence of isometric embedding equation,Gauss-Codazzi equations and Darboux equation.Meanwhile,we prove the infinitesimal rigidity of hypersurfaces in three dimensional hyperbolic and spherical space.It's based on the infinitesimal rigidity of hypersurfaces in three dimensional Euclidean space.We use Beltrami map to extend the infinitesimal rigidity in Euclidean space to space forms.Then the infinitesimal rigidity of hypersurfaces in three dimensional space forms will be proved.Finally,we prove the infinitesimal rigidity of hypersurfaces in high dimensional space forms.We point out that the previous result can also be generalized into the hypersurfaces in high dimensional space forms.It was proved by Dajczer-Rodriguez[2]firstly.As we know we can get more algebraic relations from Gauss equations.And the convexity of surface can be weakened to the condition that the rank of the second fundamental form matrix is equal to or great than two.In the paper,the infinitesimal rigidity of hypersurfaces in high dimensional Euclidean space will be proved.