We construct a family of constant curvature metrics on the Moyal plane and compute the Gauss–Bonnet term for each of them. They arise from the conformal rescaling of the metric in the orthonormal frame approach. We find a particular solution, which corresponds to the Fubini–Study metric and which equips the Moyal algebra with the geometry of a noncommutative sphere.
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Note that the rescaled frames are no longer derivations, however, if h is taken from the commutant of the algebra (or its multiplier), ei will be derivations from the Moyal algebra into the algebra of bounded operators on the Hilbert space.6 Since this does not change anything in the computations, for the sake of simplicity we work with h from the algebra itself.
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