Higher-dimensional fixed-geometry embedding for evolving shapes

Determine whether, in the higher-dimensional case, an embedding exists in which the higher-dimensional shape \(\mathcal{N}\) has fixed geometry and the evolving shapes \(N\) are higher-dimensional analogues of conic sections.

Background

The paper interprets the evolution of a transported shape $N$ through an enlarged $SL(m+1,\mathbb{R})$ orbit, represented by a higher-dimensional shape $$\mathcal{N}=SL(m+1,\mathbb{R})/\mathrm{Stab}(\vec n)$$. In the explicitly analyzed three-dimensional embedding picture, $\mathcal{N}$ is a cone and the evolving shapes $N$ arise as conic sections whose forms change during transport.

The unresolved issue is whether this geometric interpretation extends beyond the low-dimensional example: specifically, whether one can realize the higher-dimensional analogue using a single ambient shape of fixed geometry whose intersections with suitable subspaces generate the evolving transported shapes.