Curvature and geometric modules of noncommutative spheres and tori

Abstract

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

References in corpus (5)

• Levi-Civita's Theorem for Noncommutative Tori
• Riemannian Geometry of Noncommutative Surfaces
• Pseudo-Riemannian geometry in terms of multi-linear brackets
• Multi linear formulation of differential geometry and matrix regularizations
• On the classical geometry of embedded manifolds in terms of Nambu brackets

Cited by in corpus (3)

• On the Gauss-Chern-Bonnet theorem for the noncommutative 4-sphere
• Pseudo-Riemannian geometry in terms of multi-linear brackets
• A noncommutative catenoid