Extension of the rigidity theorem to higher-dimensional ruled cones

Establish whether the rigidity theorem for isotrivial cone structures extends to arbitrary ruling dimension m and ambient manifold dimension n>3, including verification of the proof and construction of the required example in the generalized setting.

Background

The paper proves rigidity in two cases: curves (m=1) in projective spaces of arbitrary dimension n-1, and ruled surfaces (m=2) in P3. In both cases, generic isotrivial cone structures of equation type arising from ordinary differential equations are shown to be flat.

The authors indicate that the same conclusion is expected for arbitrary ruling dimension m and n>3. They note that the existing proof strategy appears applicable, but the details—especially constructing an example needed to prove genericity—have not been verified.

References

We expect that our main theorem holds for any $m$ and $n>3$. In fact, the line of proof presented here in the case $m=1$, as well as for $m=2$ with $n=4$, seems to be valid in a more general context as well, but the details (in particular, the construction of an example needed in the proof) should be carefully checked.

— Rigidity theorems for cone structures  (2608.12907 - Frelik et al., 13 Aug 2026) in Remark following the proof of Theorem 2, Section 4.2 ("Structures modeled on surfaces in P^3")