Gunther’s proof of Nash’s isometric embedding theorem
Around 1987 a German mathematician named Matthias Gunther found a new way of obtaining the existence of isometric embeddings of a Riemannian manifold. His proof appeared in [1, 2]. His approach avoids the so-called Nash-Moser iteration scheme and, therefore, the need to prove smooth tame or Moser-type estimates for the inverse of the linearized operator. This simplifies the proof of Nash’s isometric embedding theorem considerably.
This is an informal expository note describing his proof. It was originally written, because when I first learned Gunther’s proof, it had not appeared either in preprint or published form, and I felt that everyone should know about it. Moreover, since he is at Leipzig, which at the time was part of East Germany, very few mathematicians in the U.S. knew about him or his proof.
Since many still seem to be unaware of Gunther’s proof, even after he gave a talk at the International Congress of Mathematicians at Kyoto in 1990 and published his proof in the proceedings , I have updated this note and continue to distribute it. I do, however, encourage you to seek out Gunther’s own presentations of his proof.
Let be a smooth –dimensional manifold. Given an embedding , the standard inner product on induces a Riemannian metric on . We shall denote this metric by . In particular, given a Riemannian metric on , we say that the embedding is isometric , if
Let . A immersion is free if for every ,
span a –dimensional linear subspace of .
The only place where Gunther’s proof differs from earlier proofs of existence lies in showing that given a smooth, free embedding and a smooth Riemannian metric sufficiently close (in a sense to be made precise later) to , there exists a smooth embedding close to such that
Although it is not necessary, we shall simplify the exposition by assuming the existence of “global” co–ordinates on . If is compact, this is obtained by embedding smoothly into a torus of larger dimension and extending smoothly the embedding and the metric to the torus so that remains close to . Otherwise, if all we are trying to prove is a local existence theorem, we can assume that is diffeomorphic to an open set in . In the discussion below, are assumed to be global co–ordinates on . (If does not have global co–ordinates, then all the calculations below should be done using a fixed smooth background metric , instead of the flat metric implied by the global co–ordinates, and its Levi–Civita connection. Extra terms involving the curvature of and the covariant derivative of curvature appear, but they are all of lower order and do not affect the proof at all.)
Let and . For convenience we shall denote
Then ( 1) is equivalent to
Applying the standard “integration by parts” trick, ( 2) can be rewritten as
This can be written abstractly in the following form:
where is a linear operator and is bilinear. Nash’s trick, when , was to observe that the linear differential operator could be inverted by a zeroth order differential operator . More recently, M. Gromov and Bryant-Griffiths-Yang have found cases where and admits a right inverse which “loses” a fixed number of derivatives. In all cases there is a loss in regularity, so that standard implicit function theorems or contraction map arguments do not seem to apply. Instead, the so–called Nash–Moser iteration scheme must be used.
Gunther’s ingenious trick can be decribed as follows: He finds new nonlocal bilinear operators and such that
where is zeroth order and is of any given negative order, i.e. it is a bilinear smoothing operator. Actually, in the specific situation here, the operator will be identically zero. Then the contraction mapping argument can be applied to the equation
The splitting is obtained as follows: Let
Then is an invertible elliptic operator on . Apply it to both sides of ( 3). Rearranging the terms and then applying to the resulting equation, we obtain;
Since is free, there exists a unique -valued bilinear operator such that and . The isometric embedding equation now becomes
Given , define , where for every , is the unique vector lying in the span of , , satisfying the following equations
Clearly, is a right inverse for . Therefore, to solve ( 3), it suffices to solve the following:
Define . If , , is sufficiently small, then is a contraction mapping on a neighborhood of . Moreover, the linear operator is an elliptic zeroth order operator and therefore if is , , then so is . In particular, if is smooth, so is .
We have therefore obtained the following:
- 1 Matthias Gunther, On the perturbation problem associated to isometric embeddings of Riemannian manifolds, Annals of Global Analysis and Geometry 7 (1989), 69–77.
- 2 by same author, Isometric embeddings of Riemannian manifolds, Proceedings of the International Congress of Mathematicians (Kyoto, 1990), Mathematical Society of Japan, 1991, pp. 1137–1143.
- 3 John Nash, The imbedding problem for Riemannian manifolds, Annals of Mathematics 63 (1956), 20–63.