Isometric bending discriminates intrinsic from extrinsic curvature coupling of nematic order in skin

Abstract

Oriented structures in skin — hair and feather follicle axes, collagen bundles, elongatedepithelial cells — can be coarse-grained into a tangential nematic order parameter on acurved surface. Such a field couples to geometry in two physically distinct ways: intrinsically,through parallel transport and Gaussian curvature, and extrinsically, through the secondfundamental form of the embedding. The two are routinely conflated because on mostanatomical surfaces they vary together. We show that they can be separated exactly byan isometric deformation. Bending a flat patch about a single axis onto a cylinder leavesthe intrinsic geometry unchanged — the Gaussian curvature and the spin connection vanishidentically throughout the family — while the traceless second fundamental form growsin proportion to the bending curvature. The intrinsic sector is therefore rigorously blindto this deformation, and any orientational response is attributable to extrinsic couplingalone. For a patch with strong anchoring along the bending axis, we find a Fréedericksz-likereorientation threshold at a critical bending radius 𝑅𝑐 = 𝑊 2/(𝜋2ℓ), where 𝑊 is the patchwidth and ℓ = 𝐿𝑆/𝜆𝑏 is a length set by the ratio of orientational elasticity to extrinsiccoupling. The post-threshold branch is obtained in closed form through a complete ellipticintegral, and confirmed by relaxation of the full Landau–de Gennes functional with variableorder amplitude. Linear and quadratic extrinsic couplings, which differ in their behaviourunder reversal of the surface normal, share a common threshold but are separated by threeindependent observables: the sense of bending for which the transition occurs, the saturationangle (90∘ versus 45∘), and the scaling of 𝑅𝑐 with patch width (𝑊 2 versus 𝑊 ). Because 𝑅𝑐grows quadratically with 𝑊 , the threshold can be brought into an experimentally accessiblerange by widening the specimen. The experiment has a definite null outcome, which boundsthe extrinsic coupling from above.