Sharp Dimensions for Totally Skew Embeddings

Definitions and mathematical statement

For a $C^1$ embedding $f:M^d\hookrightarrow{\mathbb{R}}^N$, let $E_p=df_p(T_pM)$. The embedding is totally skew if, for every pair of distinct points $p,q\in M$, every tangent line at $f(p)$ and every tangent line at $f(q)$ are disjoint and nonparallel. Equivalently, \[ \dim\left(E_p+E_q+\operatorname{span}\{f(q)-f(p)\}\right)=2d+1. \] Thus the two tangent direction spaces and the connecting direction must be linearly independent as subspaces; disjointness of tangent spaces alone is not the full condition. Define \[ N_{\mathrm{skew}}(M)=\min\{N:\text{there is a totally skew embedding }M\hookrightarrow{\mathbb{R}}^N\}, \] with value infinity if the set is empty. The general source question asks whether the cited lower bounds for $N_{\mathrm{skew}}(M)$ are sharp. No particular lower-bound formula or range of manifolds is reproduced, so this part remains a source-dependent sharpness programme.

The explicit test case is whether there exists a totally skew embedding \[ f:D^3\hookrightarrow{\mathbb{R}}^7. \] Using the closed-disk convention $D^3=\{x\in{\mathbb{R}}^3:\|x\|\leq1\}$, this asks for an embedding differentiable up to the boundary such that for every distinct $p,q\in D^3$, \[ f(q)-f(p),\ \partial_1f(p),\ \partial_2f(p),\ \partial_3f(p),\ \partial_1f(q),\ \partial_2f(q),\ \partial_3f(q) \] are seven linearly independent vectors in ${\mathbb{R}}^7$. Boundary tangent spaces have dimension three. The source does not specify the regularity class or whether disk denotes a closed disk, open disk, or local embedded disk; these conventions require confirmation, and the displayed test case makes its convention explicit.

The supplied literature assessment checked 2026-08-21 says that the totally skew three-disc in ${\mathbb{R}}^7$ remains listed as an unresolved case in later treatments of the Ghomi--Tabachnikov bounds.

Short English statement

Ask whether known dimension lower bounds for totally skew embeddings are attained, particularly whether a three-dimensional disk embeds in ${\mathbb{R}}^7$ with tangent lines at distinct points always disjoint and nonparallel.

Sources

• Local UnsolvedMath registry, permanent ID 30000137, catalogue entry, Sharp Dimensions for Totally Skew Embeddings.
• Geometrie (2005), Oberwolfach Reports, pp. 2491--2538, DOI https://doi.org/10.4171/owr/2004/47.
• Supplied reference labeled "totally skew embeddings paper": https://ghomi.math.gatech.edu/Papers/totallyskew.pdf; the assessment associates the bounds with Ghomi--Tabachnikov.
• Supplied statement verification reports confirmation of both the sharpness question and three-disk case; supplied literature triage is dated 2026-08-21.