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When considered as submanifolds of Euclidean space, the Riemannian geometryof the round sphere and the Clifford torus may be formulated in terms ofPoisson algebraic expressions involving the embedding coordinates, and acentral object is the projection operator, projecting tangent vectors in theambient space onto the tangent space of the submanifold. In this note, we pointout that there exist noncommutative analogues of these projection operators,which implies a very natural definition of noncommutative tangent spaces asparticular projective modules. These modules carry an induced connection fromEuclidean space, and we compute its scalar curvature.

Curvature and geometric modules of noncommutative spheres and tori