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.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 April 2014
 DOI:
 10.1063/1.4871175
 arXiv:
 arXiv:1308.3330
 Bibcode:
 2014JMP....55d1705A
 Keywords:

 Mathematics  Quantum Algebra;
 High Energy Physics  Theory;
 Mathematical Physics
 EPrint:
 doi:10.1063/1.4871175