Is every transitive action conformal?

Solution 1:

No. Take $G=\mathrm{SL}(n,\mathbb{R})$ for $n \ge 3$ and consider the action on $\mathbb{R}\mathbb{P}^{n-1}$. If the action preserved the conformal class of a Riemannian metric $\langle \cdot,\cdot\rangle$, then the stabilizer $H$ of a point $[v]$ would preserve the conformal class of the metric on the tangent space $T_{[v]}\mathbb{R}\mathbb{P}^{n-1}$. But every linear transformation of $T_{[v]}\mathbb{R}\mathbb{P}^{n-1}$ is induced by an element of $H$, so $H$ cannot preserve a conformal class when $n \ge 3$.

More generally, you can find lots of actions where the point-stabilizer is "too big" to respect a conformal class.