CHAPTER 1 Fundamental Theorems Given a smooth Riemannian manifold (M n ,g), we are interested in finding a smooth map u : Mn R9, for some positive integer q, such that (1.0.1) du du g. With u = (u1, , uq), this is equivalent to (1.0.2) (du1)2 + ... + (dw 9 ) 2 =^. We then call the map u an isometric imbedding or immersion according to whether u is an imbedding or an immersion. Sometimes, we simply say u is a realization of (Mn,g) in R9. There is also a local version of the above problem in which only a sufficiently small neighborhood of some specific point on the manifold is to be isometrically embedded in Rq. Let p G Mn be the point and N(p) C Mn a neighborhood of p in Mn. If u : N(p) R9 satisfies (1.0.1) in N(p), we say that u is a local isometric imbedding of g. Now let us examine (1.0.2) closely. Suppose in some local coordinate system the metric g is given by n (1.0.3) g = ^2 Qijdxidxj in Bx C Rn. For the local isometric embedding, we need to find u= (u\--- ,uq) :B1 ^Rq such that (1.0.4) ^2diUkdjUk = gi3J 1 i j n, in B±. fe=i n (n -f-1) There are sn = equations in (1.0.4), and sn is called the Janet dimension. In general, the dimension q of the target space should be bigger than or equal to sn, i.e., q sn. If q sn, (1.0.4) may have no solutions. Throughout this chapter, sn will always denote the Janet dimension and the dimension q of the target space always satisfies q sn. The system (1.0.4) is given in local coordinates. In fact, it is globally defined, both sides of (1.0.4) being second order covariant tensors. There are three sections in this chapter. We discuss the local isometric em- bedding of analytic Riemannian manifolds in the first section and that of smooth Riemannian manifolds in the second section. In the last section, we discuss the 3
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