Smooth actions of $Aff^+ (\mathbb R)$ on compact surfaces with no fixed   point: an elementary construction
  Any compact surface supports a continuous action of the orientation preserving affine group of the real line which is fixed point free (Lima and Plante). It is generally admitted that this action can be taken smooth although it is not easy finding references for it. Here one gives a such action. 
Francisco-Javier Turiel
math.DS