Scale and conformal invariance in curved space with variation of the metric

We consider the general form of action in curved space and apply the variation of metric to this action. We also investigate the scale and conformal invariance. Using this method we present two examples. The first example is action in d dimensions curved space–time. In that case the improvement energy momentum is traceless. The second example is the Liouville theory in two dimension, where the improvement energy-momentum tensor is not traceless. Finally, we show that the conformal and scale invariance are also satisfied by variation of metric.