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Nash embedding theorems

The Nash embedding theorems are fundamental results in differential geometry, established by John Forbes Nash Jr. in the mid-1950s, that address the isometric embedding of Riemannian manifolds into Euclidean spaces while preserving the intrinsic geometry.^[1] The first, known as the C¹ Nash–Kuiper theorem, states that any Riemannian manifold of dimension n admits a C^1 isometric embedding into \mathbb{R}^{2n+1}, and for compact manifolds, this can be achieved in \mathbb{R}^{2n}; this surprising result, independently discovered by Nikolai Kuiper, shows that low-regularity embeddings can flexibly approximate distance-shortening maps without the rigidity imposed by higher smoothness.^[1]^[2] The second, the smooth Nash embedding theorem, proves that every compact Riemannian n-manifold can be smoothly (i.e., C^\infty) isometrically embedded into \mathbb{R}^q for q \geq \max\left(\frac{n(n+5)}{2}, \frac{n(n+3)}{2} + 5\right), resolving a long-standing conjecture by affirming that the extrinsic geometry of Euclidean space is rich enough to realize any intrinsic Riemannian metric.^[3] These theorems revolutionized the study of manifolds by bridging intrinsic and extrinsic geometries, enabling the realization of abstract metrics in concrete Euclidean settings and influencing fields from general relativity to computer graphics.^[4] Nash's proofs relied on innovative iterative techniques, including a nonlinear version of the inverse function theorem for the smooth case and convex integration-like methods for the C^1 case, which were later generalized via the Nash–Moser implicit function theorem to handle loss of derivatives in high-order estimates.^[5]^[3] Subsequent simplifications, such as Michael Günther's 1988 proof of the smooth theorem using only the classical inverse function theorem, have made the results more accessible, while extensions to non-compact manifolds and analytic embeddings (Nash, 1966) further broadened their scope.^[6]^[7] The theorems underscore the flexibility of low-dimensional embeddings in C^1 regularity, contrasting with the higher-dimensional requirements for smoothness, and remain central to rigidity questions in geometry.^[8]

Background

Riemannian manifolds

A Riemannian manifold is a pair (M, g), where M is a smooth manifold and g is a Riemannian metric, defined as a smooth assignment to each point p \in M of a positive-definite inner product on the tangent space T_p M. This metric tensor g_p: T_p M \times T_p M \to \mathbb{R} is symmetric and bilinear, varying smoothly over the manifold, and it generalizes the notion of a metric in Euclidean space to abstract curved spaces.^[9] The Riemannian metric induces fundamental geometric structures on the manifold, including the length of smooth curves \gamma: [a,b] \to M given by \int_a^b \sqrt{g_{\gamma(t)}(\dot{\gamma}(t), \dot{\gamma}(t))} \, dt, which in turn defines a distance function d(p,q) = \inf \{ L(\gamma) \mid \gamma \text{ connects } p \text{ to } q \}. It also allows measurement of angles between tangent vectors at a point via the cosine formula derived from the inner product, and enables the computation of curvatures, such as sectional curvature, which quantify how the manifold deviates from flatness locally. These properties make Riemannian manifolds a cornerstone for studying smooth spaces with internal geometry. Classic examples illustrate these concepts. The Euclidean space \mathbb{R}^n with the standard metric g = \sum_{i=1}^n (dx^i)^2 forms a flat Riemannian manifold, where geodesics are straight lines and all curvatures vanish. In contrast, the 2-sphere S^2 \subset \mathbb{R}^3 endowed with the induced round metric g = d\theta^2 + \sin^2 \theta \, d\phi^2 (in spherical coordinates) is a Riemannian manifold of constant positive Gaussian curvature 1, where great circles serve as geodesics.^[9] Riemannian geometry emphasizes intrinsic properties, which are those computable using only the metric tensor and do not require embedding the manifold into an ambient Euclidean space; for instance, the Gauss-Bonnet theorem relates the total curvature of a surface to its topology purely intrinsically. This contrasts with extrinsic views that depend on such embeddings, though isometric embeddings preserve the intrinsic metric.^[10]

Embeddings and immersions

In differential geometry, an immersion of a smooth manifold M into Euclidean space \mathbb{R}^d is a smooth map f: M \to \mathbb{R}^d such that the differential df_p: T_p M \to T_{f(p)} \mathbb{R}^d is injective at every point p \in M.^[11] This condition ensures that the map preserves the tangent spaces locally, meaning that near each point, M is mapped without folding or collapsing dimensions, allowing the intrinsic geometry of M to be realized extrinsically in a higher-dimensional space. Immersions capture local properties but may permit global self-intersections, such as a figure-eight curve in the plane, where the map is locally an embedding but crosses itself overall. An embedding extends this notion by requiring the immersion to also be a homeomorphism onto its image f(M) \subset \mathbb{R}^d, meaning f is injective, continuous, and its inverse from the image to M is continuous.^[12] This global topological condition prevents self-intersections, ensuring that f(M) is a submanifold of \mathbb{R}^d diffeomorphic to M. Embeddings thus provide a faithful extrinsic realization of the manifold's topology and smooth structure, playing a crucial role in studying intrinsic properties like curvature through the ambient Euclidean geometry. For Riemannian manifolds (M, g), where g is the metric tensor as defined in the context of Riemannian geometry, an isometric immersion or embedding further preserves the metric structure. Specifically, for a map f: M \to \mathbb{R}^d, it is isometric if the pullback of the Euclidean metric on \mathbb{R}^d equals g, satisfying

\langle df(X), df(Y) \rangle = g(X, Y)

for all tangent vectors X, Y \in T_p M at each p \in M.^[13] This condition maintains lengths, angles, and distances, realizing the intrinsic Riemannian geometry extrinsically without distortion. The Whitney embedding theorem establishes fundamental dimension bounds for such realizations: every compact smooth n-manifold embeds smoothly (though not necessarily isometrically) into \mathbb{R}^{2n}. This bound is sharp for generic embeddings, as certain manifolds, like the real projective plane, require at least dimension $2n to avoid self-intersections. However, isometric embeddings generally demand higher ambient dimensions to accommodate the rigidity imposed by metric preservation, often exceeding $2n depending on the curvature and topology of M.

Historical development

Pre-Nash results

In the early 20th century, David Hilbert established a foundational limitation on isometric immersions of certain Riemannian manifolds into Euclidean space. Specifically, in 1901, he proved that there exists no complete C^2 isometric immersion of the hyperbolic plane—a surface of constant negative Gaussian curvature—into \mathbb{R}^3.^[14] This result highlighted the rigidity imposed by low-dimensional ambient spaces and motivated further investigations into conditions under which isometric embeddings are possible. Hermann Weyl advanced the study in the 1910s by posing a key problem concerning the realization of metrics on the sphere. Weyl's problem asked whether a given metric on S^2 with nonnegative Gaussian curvature admits a local isometric embedding into \mathbb{R}^3, assuming analyticity of the metric.^[14] Early solutions under analytic assumptions confirmed local existence for such surfaces, though global realizations remained challenging and often required additional regularity conditions.^[15] In the 1920s and 1930s, Élie Cartan contributed significantly to both local and global aspects of isometric immersions for surfaces. Using his theory of moving frames, Cartan established global isometric immersions into \mathbb{R}^3 for certain classes of surfaces, such as those with constant positive curvature, under suitable analyticity and convexity assumptions.^[16] However, his results also underscored rigidity phenomena: in higher codimensions, such immersions often preserve the intrinsic geometry rigidly, limiting flexibility for arbitrary metrics.^[14] Maurice Janet's theorem from the 1920s addressed local embeddings for hypersurfaces specifically. Janet proved that any real-analytic hypersurface metric in \mathbb{R}^n admits a local C^\infty isometric embedding into \mathbb{R}^{n+1}.^[16] This was later generalized by Cartan in 1927 to local analytic isometric embeddings of arbitrary n-dimensional real-analytic Riemannian manifolds into \mathbb{R}^{n(n+1)/2}.^[17] These pre-Nash results were predominantly local, restricted to analytic metrics, or confined to low dimensions and specific curvature conditions, revealing significant gaps in achieving global smooth isometric embeddings for arbitrary Riemannian manifolds.^[14] For instance, complete manifolds with negative curvature resisted immersion into \mathbb{R}^3, and even some smooth metrics with sign-changing curvature defied local C^2 realizations in \mathbb{R}^3.^[14] By the mid-20th century, mathematicians recognized that increasing the dimension of the ambient Euclidean space could introduce the necessary flexibility to overcome these obstructions, paving the way for more general existence theorems.^[16]

Nash's contributions

In 1954, John Nash published a groundbreaking paper introducing the C¹ isometric embedding theorem, which asserts that any compact Riemannian manifold of dimension n can be isometrically embedded into Euclidean space \mathbb{R}^{2n} with a continuously differentiable map.^[1] This result resolved a longstanding open problem in differential geometry by showing that low-regularity isometric embeddings are possible even in dimensions where smooth ones are not, using innovative convex integration techniques to approximate and correct embeddings iteratively.^[1] Building on this, Nash's 1956 paper provided a proof of the smooth (C∞) isometric embedding theorem, demonstrating that every compact Riemannian n-manifold admits a smooth isometric embedding into a sufficiently high-dimensional Euclidean space. The construction relied on a sophisticated iteration scheme involving smoothing operators and perturbative corrections to handle the nonlinear partial differential equations arising from the isometric condition. Subsequent improvements, including those by Kuiper and others, refined the dimension estimates; the current best bound for embeddings is d \geq \max\left(\frac{n(n+5)}{2}, \frac{n(n+3)}{2} + 5\right), achieved by Michael Günther in 1988.^[7]^[3] Shortly after Nash's 1954 result, Nicolaas Kuiper extended it in 1955 to C¹ isometric immersions of open (non-compact) Riemannian manifolds into \mathbb{R}^{2n}, with embeddings possible into \mathbb{R}^{2n+1}; notably, for surfaces (n=2), immersions into \mathbb{R}^{3} are achievable.^[2] A crucial tool in Nash's 1956 proof was an early version of the Nash-Moser inverse function theorem, which Nash developed to manage the loss of derivatives in solving nonlinear elliptic PDEs through tame estimates and iterative linearization; this was later generalized by Jürgen Moser into the full Nash-Moser framework.^[18] Nash's embedding theorems earned him the American Mathematical Society's 1999 Leroy P. Steele Prize for Seminal Contribution to Research, recognizing the 1956 paper's profound influence.^[19] Furthermore, the theorems laid foundational groundwork for the h-principle in topology and geometry, as elaborated by Gromov, enabling flexible constructions of solutions to underdetermined PDE systems beyond rigid analytic constraints.^[4]

The C¹ Nash–Kuiper theorem

Statement

The C¹ Nash–Kuiper theorem states that every Riemannian manifold (M, g) of dimension n admits a C^1 isometric embedding into Euclidean space \mathbb{R}^{2n+1}, and for compact manifolds, this can be achieved in \mathbb{R}^{2n}.^[1]^[2] Unlike local embedding results, which guarantee isometric embeddings only in sufficiently small neighborhoods of points, this theorem provides a global embedding of the entire manifold that exactly preserves the Riemannian metric g. The theorem applies to manifolds with C^\infty metrics but yields C^1 regularity, highlighting the flexibility at low differentiability. A key example is the C^1 isometric immersion of the round sphere S^2 into \mathbb{R}^3, which can be "crumpled" into a small ball while preserving lengths, demonstrating the lack of rigidity in C^1 category.^[20] For the immersion variant, the required dimension can be as low as \mathbb{R}^{n+1} (codimension 1) for manifolds that admit an initial immersion into that space, as extended by Kuiper; embeddings generally require the higher dimensions mentioned to avoid self-intersections.^[2]

Key ideas in the proof

The proof of the C¹ Nash–Kuiper theorem begins by reducing the global isometric immersion problem to a series of local corrections. Given an approximate C¹ immersion of a Riemannian manifold into Euclidean space, the method starts with an initial map that nearly preserves the metric and then refines it locally on coordinate charts using a partition of unity to glue the corrections together globally. This local approach leverages short-scale oscillations to adjust the immersion without affecting distant regions, ensuring the overall map remains C¹ while progressively reducing the metric error.^[20] Central to the proof is the technique of convex integration, introduced by Nash, which constructs solutions to underdetermined partial differential equations by iteratively averaging over high-frequency perturbations. In this context, the isometric immersion equations are satisfied in a weak sense through a sequence of maps where each step solves a relaxed problem involving the convex hull of possible metric tensors. Specifically, perturbations are chosen such that their average lies in the convex set of metrics close to the target, allowing the error in the induced metric to decrease geometrically while maintaining control over the C¹ norms. This method exploits the flexibility of C¹ regularity, where rapid oscillations can approximate nonlinear constraints without enforcing higher derivatives.^[20]^[21] The wrinkling procedure forms the iterative core of the construction, where small, high-frequency "wrinkles" are added to the immersion to correct discrepancies in the second fundamental form. At each iteration, an oscillatory correction of the form W(x) = \frac{1}{\lambda} a(x) (\sin(\lambda x \cdot \xi) \zeta(x) + \cos(\lambda x \cdot \xi) \eta(x)), with \lambda \to \infty, is superimposed on the current map, where a, \zeta, \eta are smooth cutoff functions and \xi is a direction vector. These wrinkles adjust the Gaussian curvature and mean curvature locally while preserving the C¹ norm of the immersion, as the high oscillation frequency confines the derivative growth. The amplitude of the wrinkles diminishes at each step, ensuring the metric error is halved or better.^[20] To maintain positive definiteness of the induced metric throughout the iterations, a key simplification involves convex combinations of the current approximate metric with a nearby Euclidean one, ensuring the combined tensor remains strictly positive definite. This averaging step, akin to techniques in later refinements, prevents degeneracy and allows the perturbations to accumulate without violating the Riemannian structure.^[20] Convergence to a C¹ isometric immersion is achieved through estimates on the derivatives of the iterative maps, where the C⁰ norm of the perturbations is controlled by \delta_q^{1/2} and the frequency \lambda_q grows exponentially. The total metric error sums to less than any prescribed \epsilon > 0, and since higher derivatives are not controlled, the limit map is exactly C¹ but not smoother, avoiding losses in regularity beyond this order.^[20]^[21] The role of the ambient dimension is crucial: for an n-dimensional manifold, immersion into \mathbb{R}^{n+1} (codimension 1) suffices due to the extra direction accommodating the transverse oscillations of the wrinkles without forcing self-intersections or higher codimension requirements, as Kuiper extended Nash's original result for codimension at least 2. This low-dimensional flexibility highlights the theorem's contrast with smoother cases, where greater room is needed to resolve rigidity.^[20]^[22]

The smooth Nash embedding theorem

Statement

The smooth Nash embedding theorem states that every compact smooth Riemannian manifold (M, g) of dimension n admits a C^\infty isometric embedding into Euclidean space \mathbb{R}^d, where d \geq \max\left( \frac{n(n+5)}{2}, \frac{n(n+3)}{2} + 5 \right). Unlike local embedding results, which guarantee isometric embeddings only in sufficiently small neighborhoods of points, this theorem provides a global embedding of the entire manifold that exactly preserves the Riemannian metric g. The theorem assumes compactness of M, which ensures the existence of a finite-energy embedding, and smoothness of the metric g. A key example is the smooth isometric embedding of any compact Riemannian surface into \mathbb{R}^{10}, which generalizes the global isometric embedding problem for abstract metrics on surfaces beyond the specific case of Weyl's problem (positive curvature metrics on S^2 into \mathbb{R}^3, solved by Nirenberg in 1953). For the immersion variant, the required dimension is \frac{n(n+5)}{2}, though embeddings demand the higher dimension \max\left( \frac{n(n+5)}{2}, \frac{n(n+3)}{2} + 5 \right) to prevent self-intersections.

Proof overview

The proof of the smooth Nash embedding theorem begins by reducing the problem to the case of embedding a torus. For a compact Riemannian manifold M^n with metric g, one covers M with coordinate charts and extends the metric locally, then glues these via an immersion into a high-dimensional torus T^{k} with a flat metric, ensuring the immersion satisfies the isometric condition |df|^2 = g through a partial differential equation solved iteratively. This setup leverages the compactness of the torus to control global behavior while handling local isometric immersions. Central to the proof is the Nash-Moser inverse function theorem, which addresses the loss of derivatives in nonlinear elliptic partial differential equations arising from the isometric condition. Unlike the standard implicit function theorem, which fails due to derivative loss in high-order terms, the Nash-Moser version operates in Fréchet spaces of smooth functions and uses tame estimates to bound norms of higher derivatives. It handles the nonlinearity by applying smoothing operators that tame the loss, enabling solvability in spaces where classical methods break down. The iteration proceeds by starting with an approximate immersion and successively adding corrections. Each step applies the implicit function theorem locally on smoothed versions of the equation, solving for perturbations that reduce the error in the metric pullback while controlling the growth of higher derivatives through quadratic estimates and frequency decomposition. Convergence is achieved after infinitely many steps, yielding a smooth isometric immersion. To facilitate the iteration, the construction employs a free map into a high-dimensional Euclidean space \mathbb{R}^N, where N is chosen large enough to provide excess degrees of freedom. A free immersion ensures that the second fundamental form has full rank, allowing flexible adjustments without singularities, parameterized by additional coordinates beyond those needed for the metric. Subsequent works by Gromov (1986) and Günther (1988) improved the dimension to \max\left( \frac{n(n+5)}{2}, \frac{n(n+3)}{2} + 5 \right), with Günther using only the classical inverse function theorem.^[7] The required ambient dimension N is justified by the structure of the Riemannian metric, which has n(n+1)/2 independent components, necessitating at least this many parameters in the embedding map for the isometric condition, plus extra dimensions for stability and to avoid obstructions in the iteration. Nash originally took N = \frac{n(3n+11)}{2}, sufficient to embed any compact n-manifold smoothly and isometrically. In 1986, Gromov provided a significant simplification using the h-principle, demonstrating that the smooth isometric embedding problem satisfies an openness condition in the space of smooth maps when the codimension is sufficiently high. This homotopy-theoretic approach shows that formal solutions (jets satisfying the metric condition) can be homotoped to actual smooth embeddings, bypassing much of the analytic iteration for high-dimensional cases.

Applications and implications

In differential geometry

The Nash embedding theorems provide a powerful framework for realizing Riemannian manifolds extrinsically as submanifolds of Euclidean space, allowing the study of intrinsic geometric properties through extrinsic tools such as the second fundamental form, which encodes how the manifold curves within the ambient space.^[4] For instance, any smooth metric on the 2-sphere S^2 can be realized as the induced metric on a smooth submanifold of \mathbb{R}^q for sufficiently large q, enabling the analysis of Gaussian curvature and other invariants via extrinsic differential geometry.^[23] This extrinsic perspective unifies intrinsic and extrinsic geometry, confirming that all compact Riemannian manifolds admit such realizations in high-dimensional Euclidean spaces, thereby impacting the classification of manifolds by demonstrating their "Euclidean" nature and influencing topological invariants like embeddability dimensions.^[3] The C^1 Nash–Kuiper theorem highlights a striking contrast between rigidity and flexibility in low-regularity embeddings, revealing that manifolds can admit highly "wiggly" C^1 isometric immersions that crumple surfaces into arbitrarily small regions without preserving higher-order smoothness.^[24] This flexibility informs theories of convex surfaces, where C^2 rigidity holds for certain embeddings (e.g., Cohn-Vossen's theorem on convex surfaces in \mathbb{R}^3), but C^1 methods allow non-rigid deformations that bypass classical obstructions like Hilbert's theorem on smooth immersions.^[20] Such constructions underscore the breakdown of rigidity in low differentiability classes, providing counterexamples to intuitive geometric constraints and enabling new insights into the local behavior of submanifolds.^[4] Extensions of Nash's ideas, notably through Gromov's h-principle, generalize the embedding results to a broad class of partial differential relations, including the h-principle for C^1 isometric immersions and fibrations, which holds when the target dimension satisfies q \geq n+1 for n-manifolds.^[4] This principle has applications to filling theorems in symplectic geometry, where C^1 approximations of isotropic maps into symplectic manifolds (Y, \omega) with \dim Y \geq 2 \dim X + 2 facilitate the construction of Lagrangian fillings and symplectic capacities.^[4] A concrete example is the C^1 isometric embedding of snippets of the hyperbolic plane into \mathbb{R}^3, achieved via convex integration techniques that "crinkle" the surface to accommodate constant negative curvature, circumventing Hilbert's prohibition on smooth global immersions.^[25]

In partial differential equations

The Nash–Moser theorem serves as a foundational tool in the analysis of nonlinear partial differential equations (PDEs), enabling the proof of local existence for solutions in spaces of smooth functions where traditional implicit function theorems fail due to derivative losses in linearizations. This iterative scheme compensates for regularity deficits by incorporating smoothing operators and tame estimates, making it applicable to quasilinear wave equations and other systems prone to such losses.^[26]^[27] A key application arises in solving the PDEs for isometric embeddings, which model overdetermined elliptic systems and demonstrate the theorem's efficacy in handling nonlinearities with limited regularity.^[16] In general relativity, the Nash–Moser framework has been employed to address the constraint equations, supporting the construction of initial data for the Einstein field equations under specific conditions. In blowup analysis for supercritical defocusing nonlinear wave systems, the Nash embedding theorem facilitates the construction of barriers or counterexamples, notably in Terence Tao's 2016 demonstration of finite-time blowup for supercritical defocusing nonlinear wave systems in high dimensions.^[28] Extensions of the inverse function theorem through Nash–Moser rely on tame Fréchet topologies for infinite-dimensional manifolds, enabling stability analyses for solutions to the incompressible Euler equations, such as local structural results for steady states. In modern contexts, Nash–Moser techniques intersect with microlocal analysis for hyperbolic PDEs, linking to the h-principle for resolving underdetermined systems via high-dimensional approximations and flexibility in solutions.^[29]

References

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BY JOHN NASH. (Received February 26, 1954). (Revised ... imbedding in Euclidean space. This paper is limited to the construction of C1 isometric imbeddings.
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May 11, 2016 · The fundamental embedding theorems show that, under reasonable assumptions, the intrinsic and extrinsic approaches give the same classes of manifolds.
[4]
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Introduction. In this note we examine the analytical part of the famous 1954 paper of John F. Nash on the isometric embedding problem [48]. Our aim is to.
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Mar 12, 2017 · If in addition the manifold is closed, then there is a C1 isometric embedding1 in R2n. Remark 2.1.6. In Nash's original paper the C0 estimate of ...
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Riemannian geometry is the study of manifolds endowed with Riemannian metrics, which are, roughly speaking, rules for measuring lengths of tangent vectors and.<|separator|>
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The award to John Nash is for his remarkable paper: “The embedding problem for Riemannian manifolds”, Ann. of Math. (2) 63 (1956), 20–63. This paper solved an ...
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The goal of this expository talk is to present a proof of the remarkable Nash(–. Kuiper) C1 embedding theorem, which states that the unit sphere S2 can be ' ...
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We give an account of the analogies between the Nash–. Kuiper C1 solutions of the isometric embedding problem and the weak solutions of the incompressible Euler ...Missing: wrinkling Günther's trick
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The classical theory of partial differential equations is rooted in physics, where equations (are assumed to) describe the laws of nature.
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Our aim is to explain recent applications of Nash's ideas in connection with the incompressible Euler equations and Onsager's famous conjecture on the energy ...