Which Riemannian manifolds admit a finite dimensional transitive Lie group action? This is a basically an adjusted version of my earlier question about how to define a convolution algebra on a general Riemannian manifold.  The motivation for asking such a question of course comes from the observation that if G is a group and X is a manifold and the action of G on X is transitive, then the pullback from each point in X to its orbit is faithful.  This then cuts out an ideal in the convolution algebra on G which would (hopefully) correspond to some type of general convolution on X (and that would be pretty handy to have for many obvious reasons).
My intuition is that for 2D surfaces (which is the case I am most interested in right now), the group is going to be something like $PSL(2) / \pi(X)$ with the action obtained by the pushforward of the action of $PSL(2)$ on $RP^2$ by the universal covering of $RP^2 / \pi(X)$.  Of course, trying to work all of this out via quotient relations is an enormous pain in the neck, so it would be nice to maybe avoid some headaches and instead try to find maybe some standard references for this sort of thing (if they exist at all).
 A: If a connected complete (edit) Riemannian manifold $M$ has an isometric transitive effective action of a Lie group $G$, then the manifold is diffeomorphic to $G/G_x$, where $G_x$ is the stabilizer of a point $x\in M$. Since $G_x$ preserves the metric at $T_x M$, then $G_x$ must be compact (the action $G_x:T_xM\to T_xM$ is faithful by considering the exponential map). Thus, one obtains all such manifolds by quotienting a Lie group by a compact subgroup.
Another viewpoint is that the Lie algebra of the identity component of $G$ gives rise to a subspace of the Killing vector fields on $M$. When you restrict the vector space of Killing fields to a point $x$, then the image should map onto the tangent space $T_x M$. I think this is necessary and sufficient. 
An interesting example is the Poincare dodecahedral space. In fact, one can decide which elliptic 3-manifolds are homogeneous, as the ones of the form $SU(2)/\Gamma$, where $\Gamma \leq SU(2)$ is finite. 
A: I think that one of the nicest results in this direction may be the one in Ambrose and Singer's paper "On homogeneous Riemannian manifolds". They give a local NASC for a complete Riemannian manifold to be homogeneous. There is a follow-up book by Tricerri and van Hecke called Homogeneous structures on Riemannian manifolds.
A: At least in the compact case, there's a topological obstruction.  In a 2005 paper, Mostow proved that a compact manifold that admits a transitive Lie group action must have nonnegative Euler characteristic.  Here's the reference:
MR2174096 (2007e:22015)
Mostow, G. D.
A structure theorem for homogeneous spaces.
Geom. Dedicata 114 (2005), 87--102. 
Of course, even if X admits a transitive Lie group action, most Riemannian metrics on X will not be homogeneous.  (You didn't say whether you wanted actions by isometries, but I assume that's what you're interested in, because otherwise the Riemannian structure on X is irrelevant.)  In the 2D case, the only compact, connected, homogeneous Riemann surfaces are the sphere, $RP^2$, the torus (and maybe the Klein bottle?)*, all with constant-curvature metrics.  In general, the group has to be compact, because the isometry group of a compact Riemannian manifold is itself compact.
*EDIT 3: An earlier paper by Mostow constructed a transitive group action on the Klein bottle, but I doubt that this action preserves a Riemannian metric.  It's a complicated construction, so I haven't had a chance to work through it in detail, but here's the reference:
Mostow, G.D., The Extensibility of Local Lie Groups of Transformations and Groups on Surfaces. Ann. Math., Second Series, (52) No. 3 (1950), 606-636.
I don't know what's known in the noncompact case.
