Temperature duality on Riemann surface and cosmological solutions for genus g = 1 and 2

A Bosonic string model at finite temperature on the gravition g(mu nu) and the dilaton phi background field is examined. Moreover, the duality relatio n of energy momentum tensor on high genus Riemann surface is derived. At th e same time, the temperature duality invariance for the action of string ga s matter is proved in 4-D Robertson-Walker metric, the string cosmological solutions and temperature duality of the equations of motion for genus g = 1 and 2 are also investigated.