Mass Generation from Embedding Geometry in Surface Nematics

We show that a nematic field constrained to a curved embedded surface develops an emergent geometric mass in its leading isotropic interaction sector. An auxiliary embedding-space closure mediated by the surface spin connection yields a massive scalar mode ${\chi }_n$ with mass set by the extrinsic curvature invariant $m^2=K_{ab}{K}^{ab}$. This mass arises directly from embedding geometry, promoting the intrinsic massless nematic interaction into a geometry-controlled massive field. The resulting theory identifies Gaussian curvature as a distributed geometric charge and establishes embedding geometry as the regulator of defect interactions on curved nematic membranes.