Lowering Günther's bound: smooth isometric immersions of arbitrary Riemannian manifolds into semi-Euclidean spaces of dimension below c(n)

Every smooth n-dimensional Riemannian manifold admits a smooth isometric embedding into a Euclidean space of dimension at most Günther's bound $c(n)=\max\{n(n+5)/2,\; n(n+3)/2+5\}$, and finding the lowest possible ambient dimension is a major open problem of isometric immersion theory. After constructing explicit smooth isometric immersions and embeddings of multiple warped product metrics into quadrics of semi-Euclidean spaces, Mirandola and Vitório posed the natural general question: does every n-dimensional Riemannian manifold admit a smooth isometric immersion into a semi-Euclidean space of some positive index (indefinite flat metric) and dimension strictly smaller than $c(n)$? Günther's bound remains the best general bound in either definite or indefinite ambient signature; low-regularity analogues ($C^{1,\theta}$ isometric embeddings into $\mathbb{R}^{2n}$) and partial non-existence results (flat normal connection, bounded mean curvature) all avoid the stated question; and the classical two-dimensional benchmark — whether the hyperbolic plane admits a $C^2$ isometric immersion into $\mathbb{R}^4$ — remains open per Aminov (2020) and Sabitov's 2023 review. The posed question is therefore open in both directions: no sub-$c(n)$ general bound is known, and no counterexample manifold is known.

Background

An isometric immersion of a Riemannian manifold $(M^n,g)$ into a flat ambient space is a smooth map $f$ whose differential preserves inner products, $\langle df(X),df(Y)\rangle=g(X,Y)$; an isometric embedding is an isometric immersion that is injective and a homeomorphism onto its image. The semi-Euclidean space $\mathbb{R}^N_a$ is $\mathbb{R}^N$ carrying the flat bilinear form of signature $(a,N-a)$ — $a$ negative and $N-a$ positive squared coordinates — so $a=0$ is ordinary Euclidean space while $a\geq 1$ gives an indefinite, Minkowski-type target. Since an immersed copy of a Riemannian manifold must have positive definite tangent spaces, any isometric immersion into $\mathbb{R}^N_a$ requires $a\leq N-n$.

A foundational theorem of immersion theory, due to Nash, refined by Gromov and Rokhlin and then by Günther, states that every smooth $n$-dimensional Riemannian manifold admits a smooth isometric embedding into some Euclidean $\mathbb{R}^N$ with $N\leq c(n)=\max\{n(n+5)/2,\; n(n+3)/2+5\}$; this is still the best general bound known. Locally, the Janet–Cartan theorem realizes every analytic $n$-dimensional metric in $\mathbb{R}^{n(n+1)/2}$, and $n(n+1)/2$ is the conjecturally optimal global dimension, so the general global bound exceeds the local optimum by roughly $2n$. Determining the minimal ambient dimension — equivalently, the lowest possible codimension — is one of the major open problems of the field. Classical rigidity shows the answer cannot be trivially small: Hilbert proved in 1901 that the complete hyperbolic plane $\mathbb{H}^2$ (Gaussian curvature $-1$) admits no $C^2$ isometric immersion into $\mathbb{R}^3$, and it is a standing open problem whether $\mathbb{H}^2$ admits a $C^2$ isometric immersion into $\mathbb{R}^4$, or an injective one into $\mathbb{R}^5$; the best explicit constructions are Blanuša's injective immersion of $\mathbb{H}^2$ into $\mathbb{R}^6$ (1955) and Rozendorn's non-injective immersion of $\mathbb{H}^2$, written as the warped metric $dt^2+e^{2t}dx^2$, into $\mathbb{R}^5$ (1960).

A complementary line of work constructs explicit low-codimension immersions for special metric classes. For multiple warped product manifolds $M^n=I\times\mathbb{R}^{n-1}$ with metrics $d\sigma^2=\rho(t)^2dt^2+\eta_1(t)^2dx_1^2+\cdots+\eta_{n-1}(t)^2dx_{n-1}^2$, where $I$ is an interval and $\rho,\eta_j$ are positive smooth warping functions scaling the flat coordinates — a class containing the hyperbolic metrics treated by Henke and the metrics considered by Azov — Mirandola and Vitório (2015) modified Blanuša's method to produce, for every index $0\leq a\leq n-1$, explicit smooth isometric immersions into $\mathbb{R}^{4n-3-2a}_a$ and explicit smooth isometric embeddings into $\mathbb{R}^{8n-7-6a}_a$, as well as into the standard quadrics (pseudo-spheres and pseudo-hyperbolic spaces) of semi-Euclidean spaces. These ambient dimensions grow only linearly in $n$, far below $c(n)$, and for $a\geq 1$ the targets are genuinely indefinite; even more starkly, hyperbolic space $\mathbb{H}^n$ is itself a hypersurface of $\mathbb{R}^{n+1}_1$. The constructions show that special metric structure and indefinite ambient signatures can both drive the required dimension far below the general bound, and they motivated the question the authors pose immediately after their main theorem: can the general bound $c(n)$ be strictly lowered for arbitrary Riemannian manifolds once indefinite semi-Euclidean targets are allowed?

Problem Statement

For each integer $n\geq 1$, let $c(n)=\max\{n(n+5)/2,\; n(n+3)/2+5\}$ be Günther's upper bound for the dimension of a Euclidean space into which every smooth $n$-dimensional Riemannian manifold isometrically embeds. The question posed by Mirandola and Vitório after their main theorem, asked for each fixed $n$, is: does every smooth $n$-dimensional Riemannian manifold $M^n$ admit a smooth ($C^\infty$) isometric immersion — a map $f\colon M^n\to\mathbb{R}^N_a$ pulling the ambient flat metric back to the given metric, $\langle df(X),df(Y)\rangle=g(X,Y)$ for all tangent vectors $X,Y$ — into a semi-Euclidean space $\mathbb{R}^N_a$ of some positive index $a\geq 1$ (a flat ambient metric of signature $(a,N-a)$, necessarily with $a\leq N-n$) whose dimension $N$ is strictly smaller than $c(n)$?

An affirmative resolution consists of a theorem showing that for each $n$ and each smooth $n$-dimensional Riemannian manifold $M^n$ — with no restriction on completeness, compactness, or curvature — there exist integers $N<c(n)$ and $a$ with $1\leq a\leq N-n$ such that $M^n$ isometrically immerses into $\mathbb{R}^{N}_{a}$. A negative resolution consists of an explicit smooth $n$-dimensional Riemannian manifold, for some $n$, that admits no smooth ($C^\infty$) isometric immersion into $\mathbb{R}^N_a$ for any pair of integers with $1\leq a\leq N-n$ and $N<c(n)$. The immersion is not required to be injective.

References

1. Heudson Mirandola and Feliciano Vitório, Global isometric embeddings of multiple warped product metrics into quadrics, Kodai Mathematical Journal 38 (2015), no. 1, 119–134, DOI: 10.2996/kmj/1426684445; preprint arXiv:1210.1812, https://arxiv.org/abs/1210.1812
2. J. Nash, The imbedding problem for Riemannian manifolds, Annals of Mathematics (2) 63 (1956), 20–63, DOI: 10.2307/1969989, https://doi.org/10.2307/1969989
3. M. L. Gromov and V. A. Rokhlin, Embeddings and immersions in Riemannian geometry, Russian Mathematical Surveys 25 (1970), no. 5, 1–57, DOI: 10.1070/rm1970v025n05abeh003801, https://doi.org/10.1070/rm1970v025n05abeh003801
4. Matthias Günther, Zum Einbettungssatz von J. Nash, Mathematische Nachrichten 144 (1989), 165–187, DOI: 10.1002/mana.19891440113, https://doi.org/10.1002/mana.19891440113
5. Matthias Günther, Isometric embeddings of Riemannian manifolds, in Proceedings of the International Congress of Mathematicians, Kyoto 1990, Vol. II, Springer-Verlag, Tokyo, 1991, pp. 1137–1143 (zbMATH 66584), https://zbmath.org/66584
6. Robert E. Greene, Isometric embeddings of Riemannian and pseudo-Riemannian manifolds, Memoirs of the American Mathematical Society, no. 97 (1970), DOI: 10.1090/memo/0097, https://doi.org/10.1090/memo/0097
7. 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
8. 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
9. È. R. Rozendorn, Realization of the metric $ds^2=du^2+f^2(u)dv^2$ in five-dimensional Euclidean space, Doklady Akademii Nauk Armyanskoi SSR 30 (1960), 197–198 (Russian; zbMATH 3173271), https://zbmath.org/3173271
10. Wolfgang Henke, Isometrische Immersionen des $n$-dim. hyperbolischen Raumes $H^n$ in $E^{4n-3}$, Manuscripta Mathematica 34 (1981), 265–278, DOI: 10.1007/BF01165540, https://doi.org/10.1007/BF01165540
11. D. G. Azov, Embedding of some classes of complete $n$-dimensional Riemannian metrics in Euclidean spaces by Blanusa's method, Moscow University Mathematics Bulletin 40 (1985), no. 5, 64–66 (Russian original: Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1985, no. 5, 72–74; zbMATH 3958184), https://zbmath.org/3958184
12. I. Kh. Sabitov, Isometric immersions of the Lobachevskii plane in $E^4$, Siberian Mathematical Journal 30 (1989), no. 5, 805–811, DOI: 10.1007/BF00971274, https://doi.org/10.1007/BF00971274
13. 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 (2007), no. 1, 10–12, DOI: 10.1134/S0001434607070024, https://doi.org/10.1134/S0001434607070024
14. Yuriy Aminov, On isometric immersions of the Lobachevsky plane into 4-dimensional Euclidean space with flat normal connection, Journal of Mathematical Physics, Analysis, Geometry 16 (2020), no. 3, 208–220, DOI: 10.15407/mag16.03.208, https://doi.org/10.15407/mag16.03.208
15. I. Kh. Sabitov, Isometric realizations of Lobachevskii plane in $\mathbb{R}^n$, $n\geq 4$, Lobachevskii Journal of Mathematics 44 (2023), no. 10, 4558–4564, DOI: 10.1134/S1995080223100323, https://doi.org/10.1134/S1995080223100323
16. Wentao Cao and László Székelyhidi Jr., On the isometric version of Whitney's strong embedding theorem, Advances in Mathematics 460 (2025), article 110040, DOI: 10.1016/j.aim.2024.110040; preprint arXiv:2306.12879, https://arxiv.org/abs/2306.12879
17. Hongda Qiu, On equivariant isometric embeddings of Riemannian manifolds with symmetries, preprint arXiv:2511.09931 (2025), https://arxiv.org/abs/2511.09931
18. Norman Zergänge, Stability analysis of isometric embeddings, preprint arXiv:1712.02766 (2017), https://arxiv.org/abs/1712.02766
19. M. Gromov, Geometric, algebraic, and analytic descendants of Nash isometric embedding theorems, Bulletin of the American Mathematical Society 54 (2017), no. 2, 173–245, DOI: 10.1090/bull/1551, https://doi.org/10.1090/bull/1551
20. Marcos Dajczer and Ruy Tojeiro, Submanifold Theory, Universitext, Springer, 2019, DOI: 10.1007/978-1-4939-9644-5, https://doi.org/10.1007/978-1-4939-9644-5