# Large and Small Conformal Groups

It's well-known that on a Riemannian manifold $(M,g)$ with dimension larger than 2, the dimension of its conformal group $\text{conf}(M,g)$ is bounded above by ${n+2\choose 2}$. A Riemannian manifold whose conformal group has the maximum possible dimension is said to have a "large" conformal group. Some important examples include semisimple Lie groups that act by fractional linear transformations on manifolds, such as $SO(n+1,1)$, the conformal group of the standard $n$-sphere, and $SO(4,2)$, the conformal group of Minkowski space.

Now let's call any Lie group of transformations on $(M,g)$ "small" if that group is a discrete group.

There are plenty of examples of Riemannian manifolds with small isometry groups. In fact there's a neat paper (Asimov, 1976) which shows that if $G_{k+1}$ is a finite group with $k+1$ elements, then there exists a Riemannian metric on the $(k-1)$-sphere such that its isometry group is isomorphic to $G_{k+1}$.

Question: Are there examples of Riemannian manifolds with small conformal groups?

## 1 Answer

By the so-called conformal Lichnerowicz conjecture (proved by Alekseevsky, Ferrand, Schoen) a manifold has either big conformal group or there exists a metric in the conformal class such that the conformal group acts by isometries of this metric. Thus, the answer to your question is the same as for the isometry group

• By the way, my answer assumes that Riemannian metrics are those which are positively definite. For metrics of other signatures, the conformal Lichnerowicz conjecture is still open and can not be used. But in the case the conformal group is compact (which is of course the case when it is finite) the conclusion of my answer still holds: there exists a metic in the conformal class such that every conformal transformation is an isometry. This metric is obtained by averedging the initial metric with respect to the action of the conformal group. – Vladimir S Matveev Sep 11 '14 at 8:54