## The imbedding problem for Riemannian manifolds.

Princeton: Princeton University Press, 1956.

First edition of Nash’s most famous work in pure mathematics, his solution of “a deep philosophical problem concerning geometry”, first posed by Bernhard Riemann, “one of the most important pieces of mathematical analysis in this century” which “has completely changed the perspective on partial differential equations” (see below).

Nash, “the most remarkable mathematician of the second half of the century” (Mikhail Gromov), is best known for his work in game theory, for which he was awarded the 1994 Nobel Prize in Economics. However, even before completing his thesis on game theory, he had become interested in the geometric objects called manifolds, which play a role in many physical problems, including cosmology. The problem Nash solved asked whether every Riemannian manifold can be isometrically embedded in Euclidean space (‘isometrically’ means bending without stretching). The Harvard mathematician Shlomo Sternberg calls the embedding problem “a deep philosophical problem concerning the foundations of geometry that virtually every mathematician – from Riemann and Hilbert to Élie Cartan and Hermann Weyl – working in the field of differential geometry has asked himself” (quoted in *A Beautiful Mind*, p. 157).

“In 1955, Nash unveiled a stunning result to a disbelieving audience at the University of Chicago. “I did this because of a bet,” he announced.

“One of his colleagues at MIT [Warren Ambrose] had, two years earlier, challenged him. “If you’re so good, why don’t you solve the embedding problem . . . ?” When Nash took up the challenge and announced that “he had solved it, modulo details,” the consensus around Cambridge was that “he is getting nowhere.” The precise question that Nash was posing—“Is it possible to embed any Riemannian manifold in a Euclidean space?”—was a challenge that had frustrated the efforts of eminent mathematicians for three-quarters of a century.

“By the early 1950s, interest had shifted to geometric objects in higher dimensions, partly because of the large role played by distorted-time and space relationships in Einstein’s theory of relativity. Embedding means presenting a given geometric object as a subset of a space of possibly higher dimension, while preserving its essential topological properties. Take, for instance, the surface of a balloon, which is two-dimensional. You cannot put it on a blackboard, which is two-dimensional, but you can make it a subset of a space of three or more dimensions.

“John Conway, the Princeton mathematician who discovered surreal numbers, calls Nash’s result “one of the most important pieces of mathematical analysis in this century.” Nash’s theorem stated that any kind of surface that embodied a special notion of smoothness could actually be embedded in a Euclidean space. He showed, essentially, that you could fold a manifold like a handkerchief without distorting it. Nobody would have expected Nash’s theorem to be true. In fact, most people who heard the result for the first time couldn’t believe it. “It took enormous courage to attack these problems,” said Paul Cohen, a mathematician who knew Nash at MIT.“After the publication of “The Imbedding Problem for Riemannian Manifolds” in the *Annals of Mathematics*, the earlier perspective on partial differential equations was completely altered. “Many of us have the power to develop existing ideas,” said Mikhail Gromov, a geometer whose work was influenced by Nash. “We follow paths prepared by others. But most of us could never produce anything comparable to what Nash produced. It’s like lightening striking. Psychologically the barrier he broke is absolutely fantastic. He has completely changed the perspective on partial differential equations. There has been some tendency in recent decades to move from harmony to chaos. Nash said that chaos was just around the corner”” (Sylvia Nasar, Introduction to *The Essential John Nash*, 2001).

The present paper is actually the second of Nash’s papers on the embedding problem. It was preceded two years earlier by ‘*C*^{1}-isometric imbeddings,’ pp. 383-396 in *Annals of Mathematics*, Vol. 60, No. 3. This paper treats the same embedding problem but only asks for a ‘continuously differentiable’ embedding rather than an ‘infinitely differentiable’ one. Although the proof in the *C*^{1} case is easier, some of the consequences are highly surprising and counter-intuitive. A copy of this paper, also in original printed wrappers, accompanies the 1956 paper.

Nash’s solution of the embedding problem in the ‘smooth’ case was extended by Jürgen Moser to give the ‘Nash-Moser implicit function theorem,’ which is now a standard technique in the study of non-linear partial differential equations and has been used by Moser to solve problems connected to the existence of periodic orbits in celestial mechanics.

For a detailed account, see Sylvia Nasar, *A Beautiful Mind* (1998), Ch. 20.

Pp. 20-63 in Annals of Mathematics, Vol. 63, No. 1, 1956. 8vo (256 x 173 mm), pp. 190, [1]. Original printed wrappers, mint condition.

Item #3536

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