For n≥2n\ge 2, p≥1p\ge 1, and curvatures c<c~c<\tilde c, does the complete hyperbolic space Hcn\mathbb{H}^n_c admit an isometric immersion into the space form Qc~n+p\mathbb{Q}^{n+p}_{\tilde c} whose normal bundle is flat? Local solutions are abundant — Cartan (1919–20) showed that local immersions of Hn\mathbb{H}^n into R2n−1\mathbb{R}^{2n-1} exist in profusion and all have flat normal bundle — but no complete example is known in any codimension or target curvature, and Dajczer–Onti–Vlachos (2021) proved that any such immersion must have a second fundamental form of exponential growth. The classical instance (c~=0\tilde c=0, p=n−1p=n-1), a conjectural higher-dimensional Hilbert theorem recorded in Yau's problem list and by Moore and Gromov, is claimed — in an as-yet unrefereed 2025 preprint by J. D. Moore — to have a negative answer (no isometric immersion of Hn\mathbb{H}^n into R2n−1\mathbb{R}^{2n-1} at all). That preprint remains unrefereed and independently unverified and covers only the Euclidean target at codimension n−1n-1, so this instance still belongs to the open question; for the general problem, the known results are necessary conditions rather than answers — by the structure theory of Dajczer–Onti–Vlachos, any such immersion (even a local one) requires codimension p≥n−1p\ge n-1 when n≥3n\ge 3, and the second fundamental form of any complete example must grow exponentially.

Background

An isometric immersion f ⁣:Mn→M~n+pf\colon M^n\to \tilde M^{n+p} is a smooth immersion whose pullback metric equals the given Riemannian metric of MnM^n, so that ff bends MnM^n extrinsically without distorting its intrinsic distances. A space form Qcm\mathbb{Q}^m_c is the complete simply connected mm-dimensional Riemannian manifold of constant sectional curvature cc: the Euclidean space Rm\mathbb{R}^m for c=0c=0, the sphere for c>0c>0, and the hyperbolic space Hcm\mathbb{H}^m_c for c<0c<0. The extrinsic bending is measured by the second fundamental form αf\alpha_f, a symmetric bilinear form with values in the normal bundle NfMN_fM (the subbundle of the ambient tangent bundle orthogonal to f∗(TM)f_*(TM)), and the normal bundle carries an induced connection whose curvature R⊥R^\perp is called the normal curvature; the immersion has flat normal bundle when R⊥R^\perp vanishes identically, which on a simply connected domain means a globally orthonormal parallel normal frame exists.

The founding result of this circle of problems is Hilbert's theorem (1901): the complete hyperbolic plane H2\mathbb{H}^2 admits no C2C^2 isometric immersion into R3\mathbb{R}^3. Élie Cartan (1919–20) extended the local theory: for n≥3n\ge 3, open subsets of Hn\mathbb{H}^n admit no isometric immersion into R2n−2\mathbb{R}^{2n-2} at all, while local immersions into R2n−1\mathbb{R}^{2n-1} exist in abundance (via Cartan–Kähler theory; they depend on n(n−1)n(n-1) functions of one variable and correspond to higher-dimensional analogues of the sine-Gordon equation) and all of them have flat normal bundle. So codimension p=n−1p=n-1 is the critical local codimension, and the natural global question — raised in Moore's 1972 work, recorded in one of Yau's problem lists, and repeated in Gromov's 2017 survey — is whether the complete Hn\mathbb{H}^n can be isometrically immersed in R2n−1\mathbb{R}^{2n-1}; non-existence was frequently conjectured. Partial confirmations of the conjecture: Xavier (1985) and Nikolayevsky (1998) proved that quotients Hn/Γ\mathbb{H}^n/\Gamma with Γ\Gamma nontrivial cannot be isometrically immersed in R2n−1\mathbb{R}^{2n-1}, and Nikolayevsky's argument shows in the flat-normal-bundle setting that a complete McnM^n_c with c<0c<0 immersed in Qc~n+p\mathbb{Q}^{n+p}_{\tilde c}, c<c~c<\tilde c, must be the simply connected Hcn\mathbb{H}^n_c; Bolotov (2007) ruled out flat-normal-bundle immersions of Hn\mathbb{H}^n into Euclidean space with bounded mean curvature. A construction of Rozendorn (whose extrinsic geometry was analyzed by Aminov, 2009) gives a complete immersion of H2\mathbb{H}^2 into R5\mathbb{R}^5 with bounded second fundamental form but non-flat normal bundle, showing that the flat-normal-bundle hypothesis is essential for all known obstructions.

Dajczer, Onti and Vlachos (2021) posed the problem in its general space-form form and proved (Theorem 1) that if a complete McnM^n_c with c<0c<0 admits an isometric immersion into Qc~n+p\mathbb{Q}^{n+p}_{\tilde c} with c<c~c<\tilde c and flat normal bundle, then Mcn=HcnM^n_c=\mathbb{H}^n_c and the norm of αf\alpha_f must grow at least exponentially: there are x0∈Mx_0\in M and k,ℓ>0k,\ell>0 with max⁡{∥αf(x)∥:x∈Dr(x0)}≥keℓr\max\{\|\alpha_f(x)\|:x\in D_r(x_0)\}\ge k e^{\ell r} for all large rr, where Dr(x0)D_r(x_0) is the closed geodesic ball of radius rr. Since the volume of hyperbolic geodesic balls itself grows exponentially, this necessary condition is not obviously contradictory, and the existence question — the subject of this problem — remains open. In a November 2025 preprint, J. D. Moore (who had himself conjectured the non-existence in R2n−1\mathbb{R}^{2n-1}) announced a proof of the Main Theorem that for every n≥2n\ge 2 there is no isometric immersion of Hn\mathbb{H}^n into R2n−1\mathbb{R}^{2n-1} whatsoever, via the Allendoerfer–Weil Gauss–Bonnet formula for Riemannian polyhedra applied to families of asymptotic polyhedra and averaged over metric connections; this preprint is unrefereed and covers only the Euclidean target at codimension n−1n-1.

Problem Statement

Let n≥2n\ge 2 and p≥1p\ge 1 be integers and let c<c~c<\tilde c be real numbers, where Hcn\mathbb{H}^n_c denotes the complete simply connected hyperbolic nn-space of constant sectional curvature cc and Qc~n+p\mathbb{Q}^{n+p}_{\tilde c} denotes the complete simply connected space form of constant sectional curvature c~\tilde c (Euclidean space if c~=0\tilde c=0, the sphere if c~>0\tilde c>0, hyperbolic space if c~<0\tilde c<0).

Does there exist an isometric immersion f ⁣:Hcn→Qc~n+pf\colon\mathbb{H}^n_c\to\mathbb{Q}^{n+p}_{\tilde c} with flat normal bundle (i.e. with vanishing curvature of the induced normal connection)? Equivalently: characterize the parameter values (n,p,c,c~)(n,p,c,\tilde c) with c<c~c<\tilde c for which such an immersion exists.

Any answer is constrained by the known necessary conditions of Dajczer–Onti–Vlachos: their structure theory forces codimension p≥n−1p\ge n-1 when n≥3n\ge 3, already for local immersions, and the second fundamental form of any complete such immersion must have exponential growth. The instance c~=0\tilde c=0, p=n−1p=n-1 (target R2n−1\mathbb{R}^{2n-1}) is the classical higher-dimensional Hilbert conjecture, claimed settled negatively — without any flatness hypothesis — by Moore's 2025 preprint; since that preprint is unrefereed and independently unverified, this instance remains part of the open question.

References

  1. M. Dajczer, C.-R. Onti, Th. Vlachos, Isometric immersions with flat normal bundle between space forms, Archiv der Mathematik 116 (2021), 577–583. DOI: 10.1007/s00013-020-01565-x (preprint arXiv:2008.03929). https://doi.org/10.1007/s00013-020-01565-x
  2. J. D. Moore, Submanifolds of Constant Negative Curvature: A Generalization of Hilbert's Theorem, preprint, arXiv:2512.00973 (2025), 46 pp. https://arxiv.org/abs/2512.00973
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