The Moyal plane is usually considered as a deformation of the flat plane, however, the algebra could equally well describe a compact noncommutative manifold. I shall demonstrate that there exists a family of metrics on the Moyal "plane" which have a constant scalar curvature thus yielding yet another quantum sphere.

I will report on a noncommutative version of the Hodge decomposition and relation to an Euler characteristic type index for a C*-dynamical system equipped with an ergodic action by a compact Lie group (after Fathizadeh and Olivier).