Abstract
We study the quantum sphere 
as a quantum Riemannian manifold in the quantum frame bundle approach. We exhibit its 2-dimensional cotangent bundle as a direct sum Ω0,1⊕Ω1,0 in a double complex. We find the natural metric, volume form, Hodge * operator, Laplace and Maxwell operators and projective module structure. We show that the q-monopole as spin connection induces a natural Levi-Civita type connection and find its Ricci curvature and q-Dirac operator 
. We find the possibility of an antisymmetric volume form quantum correction to the Ricci curvature and Lichnerowicz-type formulae for We also remark on the geometric q-Borel-Weil-Bott construction.
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Communicated by L. Takhtajan
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Majid, S. Noncommutative Riemannian and Spin Geometry of the Standard q-Sphere. Commun. Math. Phys. 256, 255–285 (2005). https://doi.org/10.1007/s00220-005-1295-8
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DOI: https://doi.org/10.1007/s00220-005-1295-8