This paper studies geometric structures on noncommutative hypersurfaces within a module-theoretic approach to noncommutative Riemannian (spin) geometry. A construction to induce differential, Riemannian and spinorial structures from a noncommutative embedding space to a noncommutative hypersurface is developed and applied to obtain noncommutative hypersurface Dirac operators. The general construction is illustrated by studying the sequence $\mathbb{ T}^{ 2}_{ \theta} \hookrightarrow \mathbb{ S}^{ 3}_{ \theta} \hookrightarrow \mathbb{ R}^{ 4}_{ \theta}$ of noncommutative hypersurface embeddings.
Categories: math.QA hep-th math-ph math.MP