Curvature and geometric information for unrestricted shape deformations

Establish whether a useful notion of curvature, and consequently encoded geometry, survives when the transported shapes are allowed to vary in the infinite-dimensional space of smooth shape deformations beyond the finite-dimensional \(SL(d+1,\mathbb{R})\) orbit.

Background

The paper proposes replacing the finite-dimensional group orbit of ellipsoids with the space N\mathcal{N} of smooth shapes, so that shape evolution may depend on the positions of individual points on the transported shape rather than only on global group parameters. In this setting, deformations supported away from the contact point are invisible to the metric-recovery procedure and could carry additional geometric information.

The authors explicitly leave unresolved whether curvature, which is central to the Cartan-geometrical interpretation, remains meaningful in this enlarged shape space and whether it can continue to encode geometry of the base manifold.

References

It is an open question as to whether a useful notion of curvature, and hence of encoded geometry, survives in this setting.

— A Cartan-geometrical perspective on torsion and non-metricity  (2609.30093 - Iosifidis et al., 24 Sep 2026) in Section 3, subsection “The scope for further generalization”