# Nash embedding theorem

The embedding theorem of Nash (after John Forbes Nash Jr. ) is a result of the mathematical branch of Riemannian geometry . He says that every Riemannian manifold isometrically in a Euclidean space of a suitable embedded can be. “Isometric” is meant in the sense of Riemannian geometry: the lengths of tangential vectors and the lengths of curves in the manifold are retained. The usual Euclidean metric of should induce the given metric of the Riemannian manifold in the embedded submanifold , so that the following applies for the embedding in local coordinates : ${\ displaystyle \ mathbb {R} ^ {n}}$${\ displaystyle n}$ ${\ displaystyle \ mathbb {R} ^ {n}}$${\ displaystyle g}$${\ displaystyle u = (u ^ {1}, ..., u ^ {n}) \ colon M \ to \ mathbb {R} ^ {n}}$

${\ displaystyle \ sum _ {l = 1} ^ {n} {\ frac {\ partial u ^ {l}} {\ partial x ^ {i}}} {\ frac {\ partial u ^ {l}} { \ partial x ^ {j}}} = g_ {ij}}$

One can therefore always think of Riemannian manifolds as submanifolds of a Euclidean space. The dimension of the Euclidean space is generally much larger than that of the Riemannian manifold.

The analogous result for ordinary differentiable manifolds is Whitney's embedding theorem , which is of a much simpler nature.

An embedding in the local real analytic case was proved by Élie Cartan and Maurice Janet in 1926 (with , where the dimension is the Riemann manifold ). Nash proved the possibility of global embedding first for differentiable embeddings in (improved by Nicolaas Kuiper ), then in the case . In the global real analytic case, Nash gave a proof in 1966. ${\ displaystyle n = {\ tfrac {m (m + 1)} {2}}}$${\ displaystyle m}$${\ displaystyle M}$${\ displaystyle C ^ {1}}$${\ displaystyle C ^ {k}}$

Nash's proof was simplified in 1989 by Matthias Günther (University of Leipzig).

This results in each case bounds for the height of the dimension of the function of the dimension of the embedded Riemannian manifold , for example, in the case by Nash and Kuiper . In case ( ), Nash showed in 1956 the existence of a global embedding for (compact manifold ) or (non-compact case). ${\ displaystyle n}$${\ displaystyle \ mathbb {R} ^ {n}}$${\ displaystyle m}$${\ displaystyle M}$${\ displaystyle C ^ {1}}$${\ displaystyle n \ geq 2m + 1}$${\ displaystyle C ^ {k}}$${\ displaystyle k \ geq 3}$${\ displaystyle n = {\ tfrac {(3m + 11) m} {2}}}$${\ displaystyle M}$${\ displaystyle n = {\ tfrac {(3m + 11) m (m + 1)} {2}}}$

In his 1956 work, Nash also laid the foundations for the Nash-Moser technique, which was widely used in the theory of nonlinear partial differential equations.

## Individual evidence

1. Cartan Sur la possibilité de plonger un espace riemannien donné dans une espace euclidien , Ann. Soc. Polon. Math., Vol. 6, 1927, pp. 1-7
2. Janet Sur la possibilité de plonger un espace riemannien donné dans une espace euclidien , Ann. Soc. Polon. Math., Vol. 5, 1926, pp. 38-43
3. Nash C1-isometric imbeddings , Annals of Mathematics, Volume 60, 1954, pp 383-396
4. Kuiper On C1-isometric imbeddings I. Nederl. Akad. Wetensch. Proc. Ser. A., Volume 58, 1955, pp. 545-556
5. Nash: The imbedding problem-for riemannian manifolds. , Annals of Mathematics, Volume 63, 1956, pp. 20-63
6. ^ Nash Analyticity of solutions of implicit function problems with analytic data , Annals of Mathematics, Volume 84, 1966, pp. 345-355. Simplified by Robert E. Greene, Howard Jacobowitz Analytic isometric embeddings , Annals of Mathematics, Volume 91, 1971, pp. 189-204
7. ^ Matthias Günther On the embedding theorem by J. Nash , Math. Nachr., Volume 144, 1989, pp. 165-187, Günther Isometric embeddings of Riemannian Manifolds , Proc. International Congress of Mathematicians, Kyoto 1990, Volume 2, pp. 1137-1143, Günther On the perturbation problem associated to isometric embeddings of Riemannian Manifolds , Annals of Global Analysis and Geometry, Volume 7, 1989, pp. 69-77, Yang Gunther ´s proof of Nash´s embedding theorem , pdf