Recently I was notified by an e-mail from Prof. R. M. Solovay of a
fault in the last part of the published paper "The Imbedding Problem
for Riemannian Manifolds" which appeared in 1956 in the Annals of
Mathematics. At first I didn't believe that he had really found
a flaw in the arguments, thinking that he had just failed to follow
the line of argument. But when I was forced to re-examine it I saw
that indeed his critique of the argument was quite accurate. There
was simply a gap in the logic of the attempted device for assuring
the avoidance of self-intersections in the embedding of a non-compact
manifold of n intrinsic dimensions in an Euclidean space of
(n+1)*(n/2)*(3*n+11) dimensions.
In principle it is not very difficult to arrange for that, in
the context of the means being used (as of the time of that original
paper). But a rigorous argument would need to be given and it would
seem that such a repair would take a different line from the scheme
that was described but which was not correct.
Of course subsequent work by others has achieved results that
need much less in terms of the number of dimensions for embeddings,
at least for a sufficiently smooth given original Riemannian manifold
for which the embedding is sought.
Enclosed here (below) is a copy of Solovay's note that drew my
attention to the fault in the originally presented argument.
////////////////////////////////////////////////////////////////
From solovay@math.berkeley.eduThu Oct 29 20:27:33 1998
Date: Tue, 16 Jun 1998 14:44:46 -0700 (PDT)
From: "Robert M. Solovay"
To: jfnj@math.Princeton.EDU
Cc: solovay@math.berkeley.edu
Subject: Your paper on imbedding Riemannian manifolds
Dear Professor Nash,
I recently have been reading your paper "The Imbedding Problem
for Riemannian Manifolds" and have discovered what seems to be an
error in your section D [in the passage entitled "Avoidance of
self-intersections"]. As a result, I do not see how to establish your
main theorem [concerning the imbedding of non-compact manifolds] with
the stated dimension bound for the target space of 1/2(3 n^3 + 7 n^2 +
11n). It is easy to make a small modification in your proof to achieve
the slightly worse bound of 1/2(3 n^3 + 7 n^2 + 11n) + (2n+1).
To describe the problem, it helps to recall the setting of
this portion of the proof. M is a non-compact n-dimensional manifold.
There is a covering of M by closed discs N_1, N_2, ... Each of the
discs N_i contains in its interior a smaller closed disc N'_i such
that the N'_i's also cover M.
The discs N_i are divided into n+1 classes. Let c(i) be the
class of the disc N_i.
Let p and q be two points that we are trying to show are not
mapped to the same point by the immersion we have constructed. Say
that p occurs in the disc N'_i. In the troublesome case q will occur
in a disc N_j [of the same class as i] with j less than i. Of course q
also occurs in some disc N'_k [of possibly some different class.] But
there is no reason that p might occur in some N_l of the same class
with l **less than** k. In this case, the steps you take will not
prevent self-intersection.
For example, it might be that the first four open sets are
distributed among classes as follows: N_0 and N_2 are in class 0; N_1
and N_3 are in class 1. p might be in N_0 and N_3; q in N_1 and
N_2. [p and q lie in no other N_i's.]
p lies in N'_3 [and no other N'_i]; q lies in N'_2 [and no
other N'_i].
The repair I have in view is similar to your construction in
Chapter C. One first imbeds M in a small region of E^{2n+1} in a
smooth fashion so that the metric induced by this embedding is
everywhere strictly smaller than the given metric on M. One then finds
an isometric imbedding for M that induces the metric which is the
difference of the given metric on M and that provided by the first
embedding. The product map is an imbedding since the first factor is,
and by construction, it achieves the desired metric.
Sincerely yours,
Bob Solovay