# How special are homogeneous spaces?

Let $$M$$ be a smooth finite dimensional manifold, how restrictive is it to require $$M$$ to admit a smooth action by a finite dimensional Lie group $$G$$?

Related questions/approaches: Of course we need $$\mathrm{dim}(G) \geq \mathrm{dim}(M)$$, are there any results relating the minimal dimension of a Lie group acting transitively to that of $$M$$, perhaps in special cases?

Going in the other direction, any criteria which easily allow to say that a given smooth manifold does not admit a transitive group action?

EDIT: in the first question I meant to write transitive smooth group action.

• In the first question, you have to specify what sort of action you want. Besides the trivial action, every manifold admits vector fields and hence non-trivial $\mathbb{R}$ actions. – alvarezpaiva May 12 at 15:55
• The answer is "extremely special". There are plenty of restrictions on these manifolds. Take a look at the $2$ and $3$ dimensional cases, for example. But there is a long history to this question, going back to the 60's. – Ryan Budney May 13 at 2:52

## 1 Answer

I suppose you want the action to be transitive as your title suggests. In this case, a classical theorem of Mostow (in 1950 for surfaces, in 2005 in general) says that for a compact homogeneous space $$M = G/H$$ the Euler characteristics is non-negative.

Mostow, G.D.: The extensibility of local Lie groups of transformations and groups on surfaces. Ann. Math. (2) 52, 606–636 (1950)

Mostow, G.D.: A structure theorem for homogeneous spaces. Geom. Dedic. 114, 87–102 (2005)