Isometry action $SO(3)\times SO(3)\to SO(3)$

Question

There exists an isometry action of $SO(3)$ over $SO(3)$?

I have been trying to prove that the natural action $SO(3)\times GL(3,\mathbb R)\to GL(3,\mathbb R)$, $(A,B)\mapsto A\cdot B$ is an isometry action respect to the metric $g=dx_{\alpha}^\beta\otimes dx_\beta^\alpha$. Unfortunately it is not an isometry action.

Any help?

Answer 1

If you are asking whether $\operatorname{SO}(3)$ can act on itself by isometries, then the answer is yes. Put any left/right invariant Riemannian metric on $\operatorname{SO}(3)$ and then the action of $\operatorname{SO}(3)$ on itself by left/right translations is by definition an action by isometries.