Surface Minkowski tensors to characterize shapes on curved surfaces

We introduce surface Minkowski tensors to characterize rotational symmetries of shapes embedded in curved surfaces. The definition is based on a modified vector transport of the shapes boundary co-normal into a reference point which accounts for the angular defect that a classical parallel transport would introduce. This modified transport can be easily implemented for general surfaces and differently defined embedded shapes, and the associated irreducible surface Minkowski tensors give rise to the classification of shapes by their normalized eigenvalues, which are introduced as shape measures following the flat-space analog. We analyze different approximations of the embedded shapes, their influence on the surface Minkowski tensors, and the stability to perturbations of the shape and the surface. The work concludes with a series of numerical experiments showing the applicability of the approach on various surfaces and shape representations and an application in biology in which the characterization of cells in a curved monolayer of cells is considered.

💡 Research Summary

This paper introduces a novel framework—surface Minkowski tensors (SMTs)—for quantifying rotational symmetries of shapes that lie on curved manifolds. Classical Minkowski tensors, widely used in flat space to capture anisotropy, rely on parallel transport of the boundary co‑normal vector to a reference point. On a curved surface this transport accumulates an angular defect proportional to the enclosed Gaussian curvature, leading to biased symmetry measures. The authors resolve this by defining a defect‑corrected transport: the usual parallel transport angle ϕ(s,t) is supplemented with a curvature‑based correction η(s,t), yielding a total rotation f(s,t)=ϕ+η. Applying this corrected transport to the co‑normal ν produces a transported vector eν(s,t) that faithfully represents the original orientation at any chosen reference point t.

The surface Minkowski tensor of rank p is then defined as
(W_{p,1}(γ(t)) = \int_{t}^{t+L}