Isometric embeddings

Abstract

This note states results extending those of Nash [2] on isometric embeddings of Riemannian manifolds in euclidean spaces; proofs and further details will be given elsewhere.Let M be a d-dimensional C 00 manifold. For convenience, we assume throughout that manifolds, whether compact or not, are connected. A metric on M is defined to be a quadratic form on the tangent bundle of M ; note that there is no assumption of nondegeneracy. We shall assume that all metrics are C 00 . A Riemannian metric on M is a metric whose restriction to the tangent space T q at a point qÇzM is positive definite, for all g£Af. A pseudo-Riemannian, or indefinite, metric is a metric whose restriction to the tangent space at each point is nondegenerate; if the nondegenerate restriction to T q has n negative eigenvalues and p positive eigenvalues, with p+n=d } the metric is said to have signature (p, n) at q. The connectedness of M implies that the signature is independent of the choice of qÇzM.

Introdução ao teorema de Nash e às Branas-mundo

“…Com a utilização de argumentos de diferenciabilidade e regularidade das funções, o teorema proposto por Nash [1,2] [16]. Dessa forma, a dimensão D do espaço ambiente para uma imersão isométrica e local de uma variedadeV n depende das funções de imersão.…”

Some isometric embedding and rigidity results for Riemannian manifolds

“…An isometric embedding fMn +RN [1] is an embedding given locally by functions ya(x) (a = 1,...,N) satisfying N ay (X) aya(X) E dx (X) = gV(X). [2] a=1 which asserts that (a) a local isometric embedding exists in the embedding dimension (Eq. 3) in the real analytic case and (b) this embedding "depends on (n -1) functions of (n -1) variables," which is the expected count when one sets up Eq.…”

“…[4] Classically, the two main existence theorems are these: (i) The Nash embedding theorem, with refinements by several people including those given in refs. [2][3][4] (1,(6)(7)(8), [5] (here, rigid means thatfis determined up to Euclidean motion by the ds2 on M).…”

“…6, 13, 14 and unpublished data). This theory gives a precise meaning to the sense in which [1] is overdetermined for r < n(n -1)/2 and, at least in principle, gives a method for determining the "integrability conditions" [11] that must be satisfied by a dS2 in order that a formal germ of isometric embedding [2] exist around each point of M. We emphasize that having a formalism that encompasses both the geometry in the problem as well as exhibiting in computable form the algebra underlying the integrability conditions is essential to the problem (note f).…”

“…Given this general setting, our argument first puts the differential system underlying the isometric embedding system [2], together with its prolongations, in a good form suitable to the particular problem (the procedure is somewhat different from that used in the existing proofs of the BurstinCartan-Janet-SchlIfly theorem). Then we combine the classical methods of Cartan (6) [k = k(r)].…”