Consider a non-compact manifold $M$.


Does there always exist a Riemannian metric on $M$ such that the isometry group is non-compact?



Possible isometry groups of open manifolds
Let $M$ be the triply punctured 2-sphere (i.e. $M=S^2-\{p\_1, p\_2, p\_3\})$); one can also take any noncompact connected surface of finite topological type as long as $\chi(M)<0$ but the proof is a bit more involved. 


Suppose that $g$ is a Riemannian metric on $M$. To simplify matters, I consider only orientation-preserving isometries. 


Then $\mathrm{Isom}(M,g)< \mathrm{Conf}(M,g)$. I claim that $\mathrm{Conf}(M,g)$, the group of conformal automorphisms of $(M,g)$, is finite. The mapping class group of $M$ (the group of self-diffeomorphisms of $M$ modulo isotopy) is finite (isomorphic to the dihedral group of order $6$); hence, it suffices to prove that if $f: (M,g)\to (M,g)$ is a conformal automorphism isotopic to the identity then $f=\mathrm{id}$. This is quite standard (most likely, you will find it in Farb-Margalit's book on the mapping class group). 


Note that $f$ is isotopic to the identity if and only if it induces an inner automorphism of $\pi\_1(M)$. 


By the uniformization theorem, $(M,g)$ is conformal to the quotient of the hyperbolic plane $H^2$ by a discrete group of isometries $\Gamma$ (isomorphic to $F\_2$, the free group on two generators). The map $f$ then lifts to an isometry $F$ of $H^2$ (a linear-fractional transformation in, say, the Poincare disk model of $H^2$) which commutes with every element of $\Gamma$ (since $f$ induces an inner automorphism of $\pi\_1(M)\cong \Gamma$). In particular, $F$ acts on the boundary circle $S^1$ of $H^2$ fixing fixed points of all elements of $\Gamma$. Since $\Gamma$ is free of rank 2, the set of fixed points of its nontrivial elements is infinite. Hence, $F$ fixes at least three points of $S^1$. Hence (being a linear-fractional transformation), $F=\mathrm{id}$; thus, $f=\mathrm{id}$. 


To conclude, every Riemannian metric on $M$ has finite group of isometries. 


Edit: I can prove the same claim for each noncompact complete hyperbolic $n$-manifold of finite volume, but a proof is more difficult. 


In general, the question can be reformulated in purely topological terms: Which smooth noncompact connected manifolds admit smooth proper actions of noncompact Lie groups? I suspect, this was studied in 1960s... 


