Curvature and geometric modules of noncommutative spheres and tori

Journal Article · Tue Apr 15 00:00:00 EDT 2014 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.4871175· OSTI ID:22250785

Arnlind, Joakim ^[1]

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1. Department of Mathematics, Linköping University, 581 83 Linköping (Sweden)

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.

• OSTI ID: 22250785
• Journal Information: Journal of Mathematical Physics, Vol. 55, Issue 4; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
COMMUTATION RELATIONS
COORDINATES
EUCLIDEAN SPACE
GEOMETRY
PROJECTION OPERATORS
SPHERES
TORI
VECTORS