The **Nash embedding theorems** (or **imbedding theorems**), named after John Forbes Nash, state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path. For instance, bending but neither stretching nor tearing a page of paper gives an isometric embedding of the page into Euclidean space because curves drawn on the page retain the same arclength however the page is bent.

The first theorem is for continuously differentiable (*C*^{1}) embeddings and the second for analytic embeddings or embeddings that are smooth of class *C ^{k}*, 3 ≤

The *C*^{1} theorem was published in 1954, the *C ^{k}*-theorem in 1956. The real analytic theorem was first treated by Nash in 1966; his argument was simplified considerably by Greene & Jacobowitz (1971). (A local version of this result was proved by Élie Cartan and Maurice Janet in the 1920s.) In the real analytic case, the smoothing operators (see below) in the Nash inverse function argument can be replaced by Cauchy estimates. Nash's proof of the

**Theorem.** Let (*M*,*g*) be a Riemannian manifold and ƒ: *M ^{m}* →

- in class
*C*^{1}, - isometric: for any two vectors
*v*,*w*∈*T*(_{x}*M*) in the tangent space at*x*∈*M*,- ,

- ε-close to ƒ:
- .

In particular, as follows from the Whitney embedding theorem, any *m*-dimensional Riemannian manifold admits an isometric *C*^{1}-embedding into an *arbitrarily small neighborhood* in 2*m*-dimensional Euclidean space.

The theorem was originally proved by John Nash with the condition *n* ≥ *m*+2 instead of *n* ≥ *m*+1 and generalized by Nicolaas Kuiper, by a relatively easy trick.

The theorem has many counterintuitive implications. For example, it follows that any closed oriented Riemannian surface can be *C*^{1} isometrically embedded into an arbitrarily small ε-ball in Euclidean 3-space (for small there is no such *C*^{2}-embedding since from the formula for the Gauss curvature an extremal point of such an embedding would have curvature ≥ ε^{−2}). And, there exist *C*^{1} isometric embeddings of the hyperbolic plane in **R**^{3}.

The technical statement appearing in Nash's original paper is as follows: if *M* is a given *m*-dimensional Riemannian manifold (analytic or of class *C ^{k}*, 3 ≤

for all vectors *u*, *v* in *T _{p}M*. This is an undetermined system of partial differential equations (PDEs).

In a later conversation with Robert M. Solovay, Nash mentioned of a fault in the original argument in deriving the sufficing value of the dimension of the embedding space for the case of non-compact manifolds.

The Nash embedding theorem is a global theorem in the sense that the whole manifold is embedded into **R**^{n}. A local embedding theorem is much simpler and can be proved using the implicit function theorem of advanced calculus in a coordinate neighborhood of the manifold. The proof of the global embedding theorem relies on Nash's far-reaching generalization of the implicit function theorem, the Nash–Moser theorem and Newton's method with postconditioning. The basic idea of Nash's solution of the embedding problem is the use of Newton's method to prove the existence of a solution to the above system of PDEs. The standard Newton's method fails to converge when applied to the system; Nash uses smoothing operators defined by convolution to make the Newton iteration converge: this is Newton's method with postconditioning. The fact that this technique furnishes a solution is in itself an existence theorem and of independent interest. There is also an older method called Kantorovich iteration that uses Newton's method directly (without the introduction of smoothing operators).

In mathematics, an **embedding** is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

In differential geometry, a **Riemannian manifold** or **Riemannian space**(*M*, *g*) is a real, smooth manifold *M* equipped with a positive-definite inner product *g*_{p} on the tangent space *T*_{p}*M* at each point *p*. A common convention is to take *g* to be smooth, which means that for any smooth coordinate chart (*U*, *x*) on *M*, the *n*^{2} functions

**Riemannian geometry** is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a *Riemannian metric*, i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.

In mathematics, an **isometry** is a distance-preserving transformation between metric spaces, usually assumed to be bijective.

In Riemannian geometry, the **scalar curvature** is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point. Specifically, the scalar curvature represents the amount by which the volume of a small geodesic ball in a Riemannian manifold deviates from that of the standard ball in Euclidean space. In two dimensions, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In more than two dimensions, however, the curvature of Riemannian manifolds involves more than one functionally independent quantity.

In mathematics, the **Chern theorem** states that the Euler-Poincaré characteristic of a closed even-dimensional Riemannian manifold is equal to the integral of a certain polynomial of its curvature form.

In mathematics, the **homotopy principle** is a very general way to solve partial differential equations (PDEs), and more generally partial differential relations (PDRs). The h-principle is good for underdetermined PDEs or PDRs, such as occur in the immersion problem, isometric immersion problem, fluid dynamics, and other areas.

This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology.

**Richard Streit Hamilton** is Davies Professor of Mathematics at Columbia University. He is known for contributions to geometric analysis and partial differential equations. He made foundational contributions to the theory of the Ricci flow and its use in the resolution of the Poincaré conjecture and geometrization conjecture in the field of geometric topology.

In mathematics, the **soul theorem** is a theorem of Riemannian geometry that largely reduces the study of complete manifolds of non-negative sectional curvature to that of the compact case. Cheeger and Gromoll proved the theorem in 1972 by generalizing a 1969 result of Gromoll and Wolfgang Meyer. The related **soul conjecture** was formulated by Gromoll and Cheeger in 1972 and proved by Grigori Perelman in 1994 with an astonishingly concise proof.

In differential geometry, the **Laplace–Beltrami operator** is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named after Pierre-Simon Laplace and Eugenio Beltrami.

In mathematics, a **manifold** is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or *n-manifold* for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to the Euclidean space of dimension n.

In mathematics, a **differentiable manifold** is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart.

In the mathematical field of analysis, the **Nash–Moser theorem**, discovered by mathematician John Forbes Nash and named for him and Jürgen Moser, is a generalization of the inverse function theorem on Banach spaces to settings when the required solution mapping for the linearized problem is not bounded.

In mathematics, **systolic geometry** is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth, and others, in its arithmetical, ergodic, and topological manifestations. See also a slower-paced Introduction to systolic geometry.

In the mathematical field of differential geometry, a smooth map from one Riemannian manifold to another Riemannian manifold is called **harmonic** if its coordinate representatives satisfy a certain nonlinear partial differential equation. This partial differential equation for a mapping also arises as the Euler-Lagrange equation of a functional generalizing the Dirichlet energy. As such, the theory of harmonic maps encompasses both the theory of unit-speed geodesics in Riemannian geometry, and the theory of harmonic functions on open subsets of Euclidean space and on Riemannian manifolds.

In mathematics, in particular in functional analysis and nonlinear analysis, it is possible to define the derivative of a function between two Fréchet spaces. This notion of differentiation, as it is Gateaux derivative between Fréchet spaces, is significantly weaker than the derivative in a Banach space, even between general topological vector spaces. Nevertheless, it is the weakest notion of differentiation for which many of the familiar theorems from calculus hold. In particular, the chain rule is true. With some additional constraints on the Fréchet spaces and functions involved, there is an analog of the inverse function theorem called the Nash–Moser inverse function theorem, having wide applications in nonlinear analysis and differential geometry.

In differential geometry, Mikhail Gromov's **filling area conjecture** asserts that the hemisphere has minimum area among the orientable surfaces that fill a closed curve of given length without introducing shortcuts between its points.

In mathematics, **Liouville's theorem**, proved by Joseph Liouville in 1850, is a rigidity theorem about conformal mappings in Euclidean space. It states that any smooth conformal mapping on a domain of **R**^{n}, where *n* > 2, can be expressed as a composition of translations, similarities, orthogonal transformations and inversions: they are Möbius transformations. This theorem severely limits the variety of possible conformal mappings in **R**^{3} and higher-dimensional spaces. By contrast, conformal mappings in **R**^{2} can be much more complicated – for example, all simply connected planar domains are conformally equivalent, by the Riemann mapping theorem.

In Riemannian geometry, the **filling radius** of a Riemannian manifold *X* is a metric invariant of *X*. It was originally introduced in 1983 by Mikhail Gromov, who used it to prove his systolic inequality for essential manifolds, vastly generalizing Loewner's torus inequality and Pu's inequality for the real projective plane, and creating systolic geometry in its modern form.

- Greene, Robert E.; Jacobowitz, Howard (1971), "Analytic Isometric Embeddings",
*Annals of Mathematics*,**93**(1): 189–204, doi:10.2307/1970760, JSTOR 1970760, MR 0283728 - Günther, Matthias (1989), "Zum Einbettungssatz von J. Nash" [On the embedding theorem of J. Nash],
*Mathematische Nachrichten*(in German),**144**: 165–187, doi:10.1002/mana.19891440113, MR 1037168 - Kuiper, Nicolaas Hendrik (1955), "On
*C*^{1}-isometric imbeddings. I",*Indagationes Mathematicae (Proceedings)*,**58**: 545–556, doi:10.1016/S1385-7258(55)50075-8, MR 0075640 CS1 maint: discouraged parameter (link) - Kuiper, Nicolaas Hendrik (1955), "On
*C*^{1}-isometric imbeddings. II",*Indagationes Mathematicae (Proceedings)*,**58**: 683–689, doi:10.1016/S1385-7258(55)50093-X, MR 0075640 CS1 maint: discouraged parameter (link) - Nash, John (1954), "
*C*^{1}-isometric imbeddings",*Annals of Mathematics*,**60**(3): 383–396, doi:10.2307/1969840, JSTOR 1969840, MR 0065993 CS1 maint: discouraged parameter (link). - Nash, John (1956), "The imbedding problem for Riemannian manifolds",
*Annals of Mathematics*,**63**(1): 20–63, doi:10.2307/1969989, JSTOR 1969989, MR 0075639 CS1 maint: discouraged parameter (link). - Nash, John (1966), "Analyticity of the solutions of implicit function problem with analytic data",
*Annals of Mathematics*,**84**(3): 345–355, doi:10.2307/1970448, JSTOR 1970448, MR 0205266 CS1 maint: discouraged parameter (link).

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