The Isometric Embedding Of S~2 In R~4

The isometric embedding problem is one of the important problems in Riemannian submanifolds geometry.With the deepening of the research object,Riemann came up with the concept of Riemannian manifold in his inaugural address in 1868,i.e.,an abstract manifold with an orthography metric.Thus,naturally there arose a question whether the Riemmannian manifold is a submanifold with an induced metric in a certain Euclidean space,this is,the existence problem of isometric embedding.In addition,there are many other properties in isometric embedding,such as openness.All of these play a key role in solving the isometric embedding process.In this thesis,we study the isometric embedding problem of S2 in R4.In the first part of this work,we mainly transform the problem of four-dimensional Euclidean space into three-dimensional warped inner product space and prove its openness.On the basis of the proof in Li-Wang[1],we add the star shaped condition and give another proof of the open theorem.It is mainly through that the solution of the linearized equation of we need.During the process of solving,we can find that the linearized equation is an elliptic equation.Therefore,we can obtain the final result by using the maximum principle.Meanwhile,we review the classical differential surface theory and discuss the iso-metric embedding of the standard flat torus and the flat Klein bottles in R4.Moreover,the proof is given.These do the preparation work for discussing the standard spheri-cal equidistant embedded in R4.At the same time,we prove the equivalence between structural equation,Gauss-Codazzi equation and Darboux equation.Finally,we solve the standard spherical related issues in R4.On the one hand,we mainly construct a new measurement and find its Gauss c urvature.Using the linearized equation of the Gauss curvature formula,we can solve the problem of the existence for the isometric embedding in R4.On the other hand,we give a follow-up study of relatedproblems in R4.For the distance function r =(?)and t = t(r),we can find that as long as we can find the t(r),the problem of isometric embedding in R4 can be solved.For details,please refer to the ideas in Guan-Lu[2].