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 σ fulfilling some suitable conditions. Correspondingly, also the definition of the Hermitian conjugate of the tensor product of two 1-forms is modified (but reduces to the standard one if σ coincides with the permutator). The metric is real with respect to such modified *-structure