Intrinsic and extrinsic Killing fields on smooth embedded surfaces

Establish whether the intrinsic Killing-field space of every general smooth, compact, connected surface without boundary embedded in \(\mathbb{R}^3\) equals the subspace induced by restrictions of ambient infinitesimal rigid motions.

Background

The paper distinguishes intrinsic Killing fields, defined as tangential vector fields with vanishing surface rate of strain, from extrinsic Killing fields obtained by restricting ambient infinitesimal rigid motions to the surface. For smooth closed surfaces with strictly positive Gaussian curvature, the paper states that infinitesimal tangential rigidity implies equality of these two spaces.

The equality is not established for arbitrary smooth compact connected surfaces embedded in R3\mathbb{R}^3. The question matters computationally because the inexpensive six-dimensional eigenvalue procedure based on ambient rigid motions identifies the full Killing space only when the intrinsic and extrinsic spaces coincide; otherwise, the intrinsic eigenvalue procedure is required.

References

To the best of our knowledge, whether $E{\mathrm{extr}=E$ holds for a general smooth compact connected surface without a boundary embedded in $\mathbb{R}3$ is {an} open question, and so one may choose to accept $E{\mathrm{extr}=E$ as an additional assumption.

— Surface Stokes-Cahn-Hilliard System: Analysis and Structure-Preserving Discretization  (2610.06552 - Olshanskii et al., 5 Oct 2026) in Section 6, subsection 6.1, “Identification of Killing vector fields”