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| id: AMR-005-0014 |
| classification: OPEN-TRIAGE |
| wording_corrected: no |
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| # AMR-005-0014 — A totally skew 3-disc in R^7 |
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| ## Problem (corrected statement if needed) |
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| Source: S. Tabachnikov, "A Baker's Dozen of Problems", Arnold Mathematical Journal 1(1), 59–67 (2015), DOI 10.1007/s40598-014-0001-3 — Problem 6, Section 12 ("Totally Skew 3-Dimensional Disc in 7-Dimensional Space?"). Verified against the [publisher HTML](https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/14-01/) and [Crossref](https://api.crossref.org/works/10.1007/s40598-014-0001-3). |
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| Original wording: |
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| > A submanifold $M^k \subset \mathbb{R}^n$ is called totally skew if, for every two distinct points $x,y \in M$, the tangent spaces at these points are in general position (i.e., their affine span has dimension $2k+1$). Clearly, a necessary condition for being totally skew is $n \geq 2k+1$. It is proved in Ghomi and Tabachnikov [2008] that if $M^k$ is a totally skew disc in $\mathbb{R}^{2k+1}$ then $k \in \{1,3,7\}$. For $k=1$, a simple example is given by the cubic curve $(t, t^2, t^3)$, $t \in \mathbb{R}$. |
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| > **Problem 6.** Is there a totally skew 3-disc in $\mathbb{R}^7$? |
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| The dataset transcription ("Does there exist a totally skew embedded 3-disc in $\mathbb{R}^7$?") is faithful to the original; no correction needed. Explicitly, an embedding $f: D^3 \to \mathbb{R}^7$ is totally skew iff for all $x \neq y$ the affine tangent 3-planes $T_x, T_y$ satisfy $\dim(T_x + T_y + \operatorname{span}(f(y)-f(x))) = 7$, equivalently (i) the direction subspaces intersect trivially and (ii) $f(y)-f(x) \notin T_x + T_y$. |
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| ## Status / Literature |
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| **Open** (as of August 2026). Verified sources: |
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| - M. Ghomi, S. Tabachnikov, "Totally skew embeddings of manifolds", Math. Z. 258(3), 499–512 (2008). DOI 10.1007/s00209-007-0182-8 (verified via Crossref); preprint arXiv:math/0307044 (verified via arXiv API). Establishes: the basic theory of totally skew embeddings; the least ambient dimension $N(M)$ satisfies $N(M) \geq 2n+1$; generic maps $M^n \to \mathbb{R}^{4n+1}$ are totally skew; and the key restriction: a totally skew $k$-disc in $\mathbb{R}^{2k+1}$ can exist only for $k \in \{1,3,7\}$. The proof relates totally skew discs to nonsingular bilinear maps and the generalized vector field problem; the restriction $k+1 \in \{2,4,8\}$ comes from Adams's solution of the Hopf invariant one problem (a totally skew disc produces data of Hopf-construction type). This leaves $k=3$ (and $k=7$) as the undecided cases — exactly Tabachnikov's Problem 6. |
| - D. Baralić, P. Đorđević, G. Stojanović, R. Živaljević, "Topological obstructions to totally skew embeddings", arXiv:1005.3709 (2010; abstract states acceptance in Trans. Amer. Math. Soc.). Establishes: obstructions to totally skew embeddings via the geometric dimension of the stable normal bundle of the configuration space $F_2(M)$; conjectures every compact $M^n$ ($n>1$) embeds totally skew in $\mathbb{R}^{4n-2\alpha(n)+1}$, $\alpha(n)$ = binary digit sum. Does not address the minimal-dimension disc problem. |
| - Z. Norfolk, "A Local Condition for Totally Skew Embeddings", arXiv:2410.20467 (2024; verified via arXiv API and HTML version). Establishes: a third-order differential condition (an analogue of nonzero torsion) guaranteeing local total skewness; an explicit cubic polynomial $\mathbb{R}^n \to \mathbb{R}^{3n}$ giving totally skew small $n$-discs in $\mathbb{R}^{3n}$; determination of $N(\mathbb{R}^n)$ for $n$ a power of 2 (Corollary 3.1.2). Crucially, the paper states that before this work $N(M)$ was known only for $\mathbb{R}^1$ ($=3$), $S^1$ ($=4$), and $\mathbb{R}^2$ ($=6$) — confirming $N(\mathbb{R}^3)$, hence the totally skew 3-disc in $\mathbb{R}^7$ question, was still open in late 2024. The power-of-2 cases do not include $n=3$. |
| - M. Harrison, "Introducing Totally Nonparallel Immersions", arXiv:1907.11312, published Adv. Math. 374 (2020) (verified via arXiv API). Studies the weaker notion (no parallel tangent lines); every $n$-manifold admits a totally nonparallel immersion in $\mathbb{R}^{4n-1}$; $TN(\mathbb{R}P^n) = 4n-1$ for $n$ a power of 2. The totally skew condition is strictly stronger, so this does not settle the disc problem. |
| - G. Stojanović, S. Tabachnikov, "Non-existence of $n$-dimensional T-embedded discs in $\mathbb{R}^{2n}$", Comment. Math. Helv. 81(4), 877–882 (2006), DOI 10.4171/CMH/78 (verified as a Crossref-deposited reference of the Ghomi–Tabachnikov paper). A non-existence result for the closely related stronger notion of T-embedded discs in even codimension. |
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| ## Work done |
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| - Fetched and read the full publisher HTML of the source list; confirmed the dataset wording matches Problem 6 verbatim. |
| - Verified every citation above against Crossref or the arXiv API (Tabachnikov 2015, Ghomi–Tabachnikov 2008, Baralić et al. 2010, Norfolk 2024, Harrison 2019/2020, Stojanović–Tabachnikov 2006). |
| - Searched for post-2015 work resolving the problem (web searches for "totally skew disc R^7", MathOverflow threads, citing papers of Ghomi–Tabachnikov). No solution or claim of solution found; the most recent paper in the area (Norfolk, Oct 2024) implicitly confirms the problem is open. |
| - Attempted the problem directly (see Result). |
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| ## Result |
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| No solution — this is a genuinely open problem. Summary of what is known and of my own analysis: |
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| 1. **The obstruction side is settled.** Ghomi–Tabachnikov prove that a totally skew $k$-disc in $\mathbb{R}^{2k+1}$ exists only if $k \in \{1,3,7\}$, via a reduction to nonsingular bilinear maps and Adams's Hopf-invariant-one theorem. For $k=3$ (and $k=7$) the obstruction vanishes: nonsingular bilinear maps $\mathbb{R}^4 \times \mathbb{R}^4 \to \mathbb{R}^7$ do exist (quaternionic Hopf construction). So the problem sits exactly at the boundary where algebraic topology gives no answer either way. |
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| 2. **My analysis of the constructive side.** Fixing a basepoint and projecting onto its normal space, a totally skew $f: D^3 \to \mathbb{R}^7$ yields a family of tangent 3-planes $\{T_x\}$ that are pairwise complementary linear subspaces; writing $T_x$ as the graph of $A_x: \mathbb{R}^3 \to \mathbb{R}^4$, one needs $A_x - A_y$ injective for all $x \neq y$ (a map of the configuration space into the Stiefel manifold $V_3(\mathbb{R}^4)$), plus the global displacement condition $f(y)-f(x) \notin T_x + T_y$. The natural first attempt, a quadratic graph $f(x) = (x, Q(x))$ with $DQ_x(v) = B(x,v)$, fails on two counts: (a) one needs a *symmetric* nonsingular bilinear $B: \mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}^4$ for parallel-tangent freeness (the elementary candidate $B(u,v) = (u \cdot v, u \times v)$ is nonsingular but not symmetric; symmetric candidates I checked by hand, e.g. symmetrized coordinate products, all turn out singular); (b) more fundamentally, a direct computation shows the affine tangent spaces of a purely quadratic graph always intersect (one solves explicitly for the intersection parameter), consistent with the known non-existence of skew branes on nondegenerate quadrics (Sha–Solomon) and of T-embedded discs (Stojanović–Tabachnikov). Higher-order (cubic) terms are therefore essential — this is exactly the role of the torsion-like third-order condition in Norfolk's 2024 local theory, which however only produces examples in codimension $\geq 2n$ (e.g. totally skew 3-discs in $\mathbb{R}^9$), not in the critical codimension $n+1$. |
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| 3. **Status of equivalent/stronger formulations.** A totally skew embedding of all of $\mathbb{R}^3$ into $\mathbb{R}^7$ would immediately give the disc by restriction; this stronger question is equally open ($N(\mathbb{R}^3) \in \{7, 8, \dots\}$ unknown; known bounds $7 \leq N(\mathbb{R}^3) \leq 9$ from the general lower bound and Norfolk's $\mathbb{R}^{3n}$ construction). |
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| ## What remains |
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| - The core question: construct a totally skew 3-disc in $\mathbb{R}^7$ (equivalently a smooth $f$ with $\det[Df_x, Df_y, f(y)-f(x)] \neq 0$ for all $x \neq y$), or prove non-existence. Both directions seem to require new ideas: the known topological obstructions are exhausted (they yield only $k \in \{1,3,7\}$), and known local/perturbative constructions lose one or two dimensions. |
| - Natural next steps: (i) try to exploit the quaternionic nonsingular bilinear map $\mathbb{R}^4 \times \mathbb{R}^4 \to \mathbb{R}^7$ as the second-order jet of a candidate embedding and control the third-order (torsion-type) term à la Norfolk in the critical codimension; (ii) investigate whether Norfolk's local condition can be satisfied by a map $\mathbb{R}^3 \to \mathbb{R}^7$ (the space of cubic polynomials modulo the discriminant is small here, so this is a concrete finite-dimensional algebraic question); (iii) the same question for $k=7$ in $\mathbb{R}^{15}$, presumably harder. |
| - Related open problem: determine $N(\mathbb{R}^n)$ for $n$ not a power of 2, in particular $N(\mathbb{R}^3)$. |
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