Finite-path transport between prescribed endpoint shapes

Determine whether transport without slipping can be achieved along a finite path for a surface that begins with a prescribed shape \(N_{1}\) and ends with a prescribed shape \(N_{2}\).

Background

The paper proves that transport without slipping can always be arranged locally under suitable non-degeneracy and signature conditions. It emphasizes, however, that these results are local and that finite paths may impose additional restrictions.

The one-dimensional shear-transport example demonstrates such a restriction: an ellipse can be transported entirely through shape evolution only over a finite distance. This motivates the unresolved global existence question for arbitrary initial and final shapes along a finite path.

References

It remains an open question as to whether transport without slipping can be achieved for a finite path of a surface with shape N_{1} at the beginning of the path and N_{2} at the end.

— A Cartan-geometrical perspective on torsion and non-metricity  (2609.30093 - Iosifidis et al., 24 Sep 2026) in Section 3, subsection “The existence of transport without slipping”