The hyperbolic plane H2\mathbb{H}^2 — the complete, simply connected surface of constant Gaussian curvature −1-1 — is known to admit no C2C^2 isometric immersion into R3\mathbb{R}^3 (Hilbert, 1901), to admit explicit C1C^1 isometric embeddings into R3\mathbb{R}^3 (Borrelli–Denis–Lazarus–Theillière–Thibert, 2023), smooth non-injective isometric immersions into R5\mathbb{R}^5 (Rozendorn, 1960), and smooth isometric embeddings into R6\mathbb{R}^6 (Blanuša, 1955). The borderline codimension-two case — whether H2\mathbb{H}^2 admits a global C2C^2 isometric immersion into R4\mathbb{R}^4 — has remained open for over a century. Niang (Afrika Matematika, 2012) proves that no such immersion can have second fundamental form of rank 1 everywhere and constructs explicit rank-2 isometric immersions of the strict half-plane domains R×(a,+∞)⊂H2\mathbb{R}\times(a,+\infty)\subset\mathbb{H}^2, a>0a>0, leaving the global C2C^2 existence question in R4\mathbb{R}^4 open as of 2026.

Background

An isometric immersion of a Riemannian manifold (M,g)(M,g) into Euclidean space RN\mathbb{R}^N is a C2C^2 map f:M→RNf:M\to\mathbb{R}^N with injective differential such that the pullback of the Euclidean metric equals gg; intrinsically, the image preserves all lengths and angles of MM, though it may self-intersect. The hyperbolic plane H2\mathbb{H}^2 is the unique complete, simply connected Riemannian surface of constant Gaussian curvature K=−1K=-1; a concrete model is the upper half-plane {(x,y):y>0}\{(x,y):y>0\} with metric g=(dx2+dy2)/y2g=(dx^2+dy^2)/y^2. The second fundamental form IIII of an immersion measures its extrinsic curvature: for each tangent vector XX it gives the ambient acceleration ∂X∂Xf\partial_X\partial_X f projected to the normal space (the orthogonal complement of the image of the differential in the ambient tangent space); for a surface in R4\mathbb{R}^4 the normal space is 2-dimensional, so IIII takes values in R2\mathbb{R}^2 and its rank at a point is 0, 1, or 2 as a vector-valued quadratic form. The Gauss equation links intrinsic and extrinsic curvature: for an orthonormal basis X,YX,Y of the tangent space, ⟨II(X,X),II(Y,Y)⟩−∣II(X,Y)∣2=K\langle II(X,X),II(Y,Y)\rangle-|II(X,Y)|^2=K; for H2→R4\mathbb{H}^2\to\mathbb{R}^4 the right side is identically −1-1, so the extrinsic curvature never degenerates. Asymptotic coordinates are local coordinates whose coordinate curves have vanishing normal curvature; they exist wherever IIII has rank 1, and in codimension one they reduce the immersion equations to the classical sine-Gordon equation uxt=sin⁡uu_{xt}=\sin u — the starting point of Hilbert's proof.

Hilbert (1901) proved that H2\mathbb{H}^2 admits no complete C2C^2-regular isometric immersion into R3\mathbb{R}^3; Efimov (1964) extended the impossibility to all complete surfaces in R3\mathbb{R}^3 with Gaussian curvature bounded above by a negative constant. These codimension-one obstructions exploit global asymptotic coordinates (a Chebyshev net, i.e. a coordinate net in which both families of coordinate curves are parametrized by arclength) and area bounds. In the opposite, flexible direction: Nash–Kuiper-type convex integration (a technique that builds solutions of underdetermined PDE systems as limits of iteratively adding high-frequency, low-amplitude corrugations) makes the C1C^1 category completely soft, with an explicit C1C^1 isometric embedding of the entire H2\mathbb{H}^2 into R3\mathbb{R}^3 with relatively compact image constructed by Borrelli–Denis–Lazarus–Theillière–Thibert (2023); and in higher codimension, Blanuša (1955) built an explicit smooth isometric embedding H2→R6\mathbb{H}^2\to\mathbb{R}^6, while Rozendorn (1960) built an explicit smooth non-injective isometric immersion H2→R5\mathbb{H}^2\to\mathbb{R}^5 (whether even an injective one exists in R5\mathbb{R}^5 is itself open, as Mirandola–Vitório note). Poznyak (1973) proved that every smooth metric on the closed 2-disk admits a smooth isometric immersion into R4\mathbb{R}^4 — a local-bounded-domain statement in precisely the target dimension, showing that dimension 4 supports the relevant local theory and that completeness of the domain is the essential difficulty.

Dimension 4 is the sharp borderline: in the smooth category the obstruction of Hilbert–Efimov is specific to codimension one, while all known smooth constructions of the complete hyperbolic plane live in codimension three (immersion, R5\mathbb{R}^5) or higher. Sabitov (1989) showed that within generalized surfaces of revolution the hyperbolic plane can be realized in R4\mathbb{R}^4 only at piecewise-analytic C0,1C^{0,1} regularity (no C1C^1 surface-of-revolution immersion exists), and his 2023 survey still records this Lipschitz object as the best known realization in R4\mathbb{R}^4. Cartan (1919–20) determined the local threshold for hyperbolic nn-spaces: Hn\mathbb{H}^n with n≥3n\ge3 admits no local isometric immersion into RN\mathbb{R}^N for N<2n−1N<2n-1, while local immersions are abundant in codimension n−1n-1; the long-conjectured impossibility in R2n−1\mathbb{R}^{2n-1} for n≥3n\ge3 (a problem listed by Yau, 1982) was proved by Moore (2025) via the Allendoerfer–Weil Gauss–Bonnet formula for Riemannian polyhedra (the extension of the classical Gauss–Bonnet theorem to polyhedra whose faces carry intrinsic metrics) — for n=2n=2 his theorem is exactly Hilbert's, and the codimension-two space R2n\mathbb{R}^{2n}, in particular R4\mathbb{R}^4 for the plane, is untouched by the method. Necessary conditions any solution must satisfy are strong: Fontenele–Xavier (2016) proved that every Lipschitz map of a Hadamard manifold (a complete, simply connected Riemannian manifold of nonpositive sectional curvature) with pinched negative curvature into any Euclidean space is a compression — it collapses some pairs of intrinsically far-apart balls of any radius into an arbitrarily small Hausdorff neighborhood — so any isometric immersion of H2\mathbb{H}^2 into R4\mathbb{R}^4 must have complicated asymptotic behavior; Dajczer–Onti–Vlachos (2020) proved that an isometric immersion of a space form of negative curvature into a space form, with flat normal bundle (vanishing normal curvature tensor) and negative extrinsic curvature, must have second fundamental form of exponential growth — a strong constraint on the flat-normal-bundle subclass, though not an obstruction. Niang (2012) contributes a rank dichotomy: for negatively curved surfaces in R4\mathbb{R}^4, having rank-1 second fundamental form everywhere is equivalent to admitting asymptotic coordinates everywhere, which forces the surface to lie locally in an ambient R3⊂R4\mathbb{R}^3\subset\mathbb{R}^4 — hence, by Hilbert's theorem, impossible for H2\mathbb{H}^2 — while explicit rank-2 immersions exist on every strict half-plane R×(a,+∞)\mathbb{R}\times(a,+\infty), a>0a>0, of H2\mathbb{H}^2.

Problem Statement

Let H2\mathbb{H}^2 be the complete, simply connected Riemannian surface of constant Gaussian curvature −1-1, realized concretely as the upper half-plane {(x,y):y>0}\{(x,y):y>0\} with metric g=(dx2+dy2)/y2g=(dx^2+dy^2)/y^2. Does there exist a global C2C^2 isometric immersion f:H2→R4f:\mathbb{H}^2\to\mathbb{R}^4, i.e. a twice continuously differentiable map, defined on all of H2\mathbb{H}^2, such that

⟨∂xf,∂xf⟩=1y2,⟨∂xf,∂yf⟩=0,⟨∂yf,∂yf⟩=1y2\langle \partial_x f,\partial_x f\rangle=\frac{1}{y^2},\qquad \langle\partial_x f,\partial_y f\rangle=0,\qquad\langle\partial_y f,\partial_y f\rangle=\frac{1}{y^2}

identically (which forces dfdf to be injective and the intrinsic metric to be exactly gg, so completeness of the induced metric is automatic)? No constraint is placed on the rank of the second fundamental form — rank 1 on part of the surface is allowed, though Niang (2012) proved rank 1 everywhere is impossible — and no injectivity is required. This is the classical codimension-two, C2C^2-regularity case of Hilbert's problem for the hyperbolic plane, exactly as left open by Niang (2012).

References

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