  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.
