I have the following question.
Let $(M,g_{ab})$ be a Riemannian manifold $M$ with metric $g$, and with an action of a Lie group $G$. Moreover, the Riemannian metric $g_{ab}$ is taken to be invariant under the action of the Lie group $G$ (i.e. the action is isometric with respect to $g_{ab}$).
Now, I perform the change of metric $g_{ab}(x) \rightarrow g'_{ab}(x) = e^{\phi(x)}g_{ab}(x)$. Is the new metric also invariant under the action of the Lie group $G$ (i.e. is the action of $G$ isometric with respect to $g'_{ab}$)?