Metrics on the Real Quantum Plane
  Using the frame formalism we determine some possible metrics and metric-compatible connections on the noncommutative differential geometry of the real quantum plane. By definition a metric maps the tensor product of two 1-forms into a `function' on the quantum plane. It is symmetric in a modified sense, namely in the definition of symmetry one has to replace the permutator map with a deformed map \sigma fulfilling some suitable conditions. Correspondingly, also the definition of the hermitean conjugate of the tensor product of two 1-forms is modified (but reduces to the standard one if \sigma coincides with the permutator). The metric is real with respect to such modified *-structure. 
G. Fiore (1 and 2), M. Maceda (3), J. Madore (3 and 4) ((1) Dip. di   Matematica e Applicazioni, Fac. di Ingegneria, (2) I.N...