The problem asks whether the hyperbolic plane H2H^2 — the complete, simply connected surface of constant Gaussian curvature −1-1, classically called the Lobachevsky plane — admits a regular (at least C2C^2) isometric immersion into Euclidean 4-space E4E^4 defined on the entire plane. By Hilbert's theorem no such immersion exists into E3E^3, while Blanuša realized H2H^2 smoothly in E6E^6 and Rozendorn in E5E^5, so E4E^4 is the unique minimal dimension left open, a question outstanding since the 1950s. The status is open: the known partial results — non-existence for rank-one second fundamental form, the flat-normal-connection and Lagrangian reductions, and local-solution analyses of the governing Gauss–Codazzi–Ricci system — leave the general existence question untouched.

Background

An isometric immersion of a Riemannian manifold is a smooth map into another Riemannian manifold that preserves the lengths of tangent vectors; for a surface it is a parametrization whose induced first fundamental form equals the given metric. "Regular" means the parametrization is sufficiently smooth (at least C2C^2) and has no singular points, i.e. its differential has rank 2 everywhere. The Lobachevsky plane, or hyperbolic plane H2H^2, is the complete, simply connected two-dimensional Riemannian manifold of constant Gaussian curvature −1-1; in global horocyclic coordinates (u,v)∈R2(u,v)\in\mathbb{R}^2 — obtained from the upper-half-plane coordinates (x,y)(x,y), y>0y>0, with metric (dx2+dy2)/y2(dx^2+dy^2)/y^2 by u=ln⁡yu=\ln y, v=xv=x — its metric is ds2=du2+e−2udv2ds^2=du^2+e^{-2u}dv^2. Completeness means every geodesic extends indefinitely, so an immersion of the entire H2H^2 is defined on a complete noncompact domain.

The classical backdrop is Hilbert's theorem (1901): no complete regular surface of constant negative Gaussian curvature can be isometrically immersed in Euclidean 3-space E3E^3; Efimov extended the obstruction to complete surfaces whose Gaussian curvature is bounded above by a negative constant. Codimension 1 is therefore rigidly impossible. In the opposite, low-regularity direction the problem is flexible: by the C1C^1 isometric embedding theorem of Nash and Kuiper, H2H^2 admits C1C^1 isometric immersions even into E3E^3, and an explicit such embedding, with relatively compact (i.e. bounded) image, was constructed in 2023 by Borrelli, Denis, Lazarus, Theillière and Thibert. For regular immersions of the whole plane the known ambient dimensions are: E3E^3 impossible (Hilbert); E5E^5 possible — Rozendorn (1960) constructed a C∞C^\infty isometric immersion of the complete hyperbolic plane in 5-dimensional Euclidean space, realizing metrics of the form ds2=du2+f2(u)dv2ds^2=du^2+f^2(u)dv^2 (the hyperbolic metric arises for f(u)=e−uf(u)=e^{-u}), a surface whose extrinsic geometry Aminov later studied in detail; and E6E^6 possible via Blanuša's 1955 smooth isometric embedding. The one remaining dimension, E4E^4, is the open case.

For a surface in E4E^4 the normal space at each point is a plane, and the compatibility conditions for an isometric immersion — the Gauss, Codazzi and Ricci equations, a PDE system linking the first fundamental form, the second fundamental form (the quadratic form measuring how the surface bends in the ambient space), and the normal connection (which measures twisting of the normal plane as one moves along the surface) — become underdetermined: the second fundamental form can have rank 2 and the normal connection can be non-flat, so the codimension-1 mechanisms behind Hilbert's and Efimov's proofs do not apply. Published partial results delineate the difficulty. Niang proved that there is no isometric immersion of H2H^2 into R4\mathbb{R}^4 whose second fundamental form has rank 1 at every point (equivalently, one whose image is locally contained in a 3-dimensional subspace), and with coauthors reformulated the general problem as a PDE system, obtaining local solutions but no complete ones under natural additional conditions. Bolotov and Brander analyzed the restricted classes — immersions with flat normal connection (the case where the curvature of the normal connection vanishes identically), and Lagrangian immersions (immersions of an nn-dimensional manifold into complex nn-space Cn\mathbb{C}^n whose tangent spaces are carried by the ambient complex structure onto the corresponding normal spaces) — that are the only settings to which a Hilbert-type non-immersion theorem could generalize, and global non-existence remains open even there. Bulca and Arslan (2013) pose this question as an open problem.

Problem Statement

Does there exist an isometric regular immersion of the entire Lobachevsky plane into Euclidean 4-space E4E^4? That is: is there a map X:H2→E4X:H^2\to E^4, defined on the complete hyperbolic plane H2H^2 of constant Gaussian curvature −1-1, that is at least C2C^2, has rank-2 differential at every point, and induces the hyperbolic metric, so that ⟨dX(p),dX(q)⟩=⟨p,q⟩\langle dX(p),dX(q)\rangle=\langle p,q\rangle for all tangent vectors p,qp,q?

Equivalently, in the global horocyclic coordinates (u,v)∈R2(u,v)\in\mathbb{R}^2 with

ds2=du2+e−2udv2,ds^2=du^2+e^{-2u}dv^2,

do there exist C2C^2 functions X1,…,X4X_1,\dots,X_4 on all of R2\mathbb{R}^2 such that the surface X(u,v)=(X1(u,v),…,X4(u,v))X(u,v)=(X_1(u,v),\dots,X_4(u,v)) satisfies

⟨Xu,Xu⟩=1,⟨Xu,Xv⟩=0,⟨Xv,Xv⟩=e−2u\langle X_u,X_u\rangle=1,\qquad \langle X_u,X_v\rangle=0,\qquad \langle X_v,X_v\rangle=e^{-2u}

at every point of R2\mathbb{R}^2, i.e. a regular isometric immersion of the whole plane with no singular points?

The question is posed in the regular category: for C1C^1 immersions the answer is positive already in E3E^3, in E5E^5 and all higher dimensions regular immersions of the entire plane exist, and E3E^3 is impossible for regular immersions; E4E^4 is the only dimension in question. A resolution is required in either direction: an explicit construction of such an immersion, or a proof that none exists.

References

  1. D. Hilbert, "Ueber Flächen von constanter Gaussscher Krümmung", Transactions of the American Mathematical Society 2 (1901), 87–99. DOI: 10.1090/s0002-9947-1901-1500557-5, https://doi.org/10.1090/s0002-9947-1901-1500557-5
  2. N. V. Efimov, "Appearance of singularities on surfaces of negative curvature" (English translation), American Mathematical Society Translations, Series 2, Volume 66 (1968), 154–190; Russian original: Matematicheskiĭ Sbornik 64(106) (1964), 286–320. DOI: 10.1090/trans2/066/07, https://doi.org/10.1090/trans2/066/07; Zbl: 0179.26605, https://zbmath.org/?q=an%3A0179.26605
  3. D. Blanuša, "Über die Einbettung hyperbolischer Räume in euklidische Räume", Monatshefte für Mathematik 59 (1955), 217–229. DOI: 10.1007/BF01303796, https://doi.org/10.1007/BF01303796
  4. È. R. Rozendorn, "Realisierung der Metrik ds^2 = du^2 + f^2(u)dv^2 im fünfdimensionalen euklidischen Raum", Doklady Akademii Nauk Armyanskoĭ SSR 30 (1960), 197–198. Zbl: 0106.14701, https://zbmath.org/?q=an%3A0106.14701
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  6. J. Nash, "C^1 isometric imbeddings", Annals of Mathematics (2) 60 (1954), 383–396. DOI: 10.2307/1969840, https://doi.org/10.2307/1969840
  7. N. H. Kuiper, "On C^1-isometric imbeddings. I", Indagationes Mathematicae (Proceedings) 58 (1955), 545–556. DOI: 10.1016/S1385-7258(55)50075-8, https://doi.org/10.1016/S1385-7258(55)50075-8
  8. D. V. Bolotov, "On isometric immersions with flat normal connection of the hyperbolic space L^n into Euclidean space E^{n+m}", Mathematical Notes 82(1–2) (2007), 10–12. DOI: 10.1134/S0001434607070024, https://doi.org/10.1134/S0001434607070024
  9. D. Brander, "Results related to generalizations of Hilbert's non-immersibility theorem for the hyperbolic plane", Electronic Research Announcements of Mathematical Sciences 15 (2008), 8–16; preprint arXiv:0710.0507 (2007, v4 2008). arXiv ID: 0710.0507, https://arxiv.org/abs/0710.0507
  10. Yu. A. Aminov, "Extrinsic geometric properties of the Rozendorn surface, an isometric immersion of the Lobachevskiĭ plane in E^5", Sbornik: Mathematics 200(11) (2009), 1575–1586. DOI: 10.1070/SM2009v200n11ABEH004051, https://doi.org/10.1070/SM2009v200n11ABEH004051
  11. A. Niang, "About the existence of isometric immersion(s) of the hyperbolic plane H^2 in R^4" (French title: "Autour de l'existence d'immersion(s) isométrique(s) du plan hyperbolique H^2 dans R^4"), Afrika Matematika 24(4) (2013), 457–464. DOI: 10.1007/s13370-012-0071-0, https://doi.org/10.1007/s13370-012-0071-0
  12. M. Dione, B. G. Diop, A. Niang, "A PDE for smooth isometric immersion of an open set of the hyperbolic plane H^2 into R^4", Journal of Mathematics 2021 (2021), Article ID 2012354. DOI: 10.1155/2021/2012354, https://doi.org/10.1155/2021/2012354
  13. V. Borrelli, R. Denis, F. Lazarus, M. Theillière, B. Thibert, "The Hyperbolic Plane in E^3", arXiv:2303.12449 (2023). arXiv ID: 2303.12449, https://arxiv.org/abs/2303.12449
  14. B. Bulca, K. Arslan, "Surfaces given with the Monge patch in E^4", arXiv:1305.4373 (2013). DOI: 10.48550/arXiv.1305.4373, https://arxiv.org/abs/1305.4373