Which smooth manifolds can be made homogeneous?

During our Riemannian geometry course, Manan asked which homogeneous smooth manifolds

can be made Riemannian homogeneous

The answer is very simple: no, the necessary and sufficient condition is that point stabilizers , which act linearly on the tangent space , must have pre-compact image in .

^1
2
3

It is easy to construct counterexamples.

Every 2-transitive Lie group action on a connected smooth manifold (of dim ?) cannot be Riemannian because it cannot preserve a metric. The action can at most take two equidistant points to another pair of equidistant points.

Also, we can easily count the dimension of the Lie group when is connected.

This means, for example, for we have .

For these reasons,

does not preserve any Riemannian metric.

Is every smooth manifold homogeneous?

• Given any two points in a connected manifold, there exists a diffeomorphism taking one to the other. ⁴
• However, not all smooth manifolds are homogeneous -spaces for (finite dim) Lie groups : there are non-trivial necessary conditions ⁵.

(Mostow, 2005) is a compact homogeneous -space

⁶

Is a quotient of a Lie-homogeneous again Lie-homogeneous?

No. By Mostow’s result, higher genus compact orientable surfaces are not Lie-homogeneous, but their universal cover is which is Lie-homogeneous.

The fact that they are not Riemannian homogeneous is easily proven by Riemann surface techniques. ⁷

Classification of connected isotropic Riemannian manifolds

Classification of every connected isotropic Riemannian manifolds is actually very simple: they are precisely all globally symmetric spaces of rank 1. These are also precisely all the Riemannian manifolds whose isometry group act transitively on all pairs of equidistant points.

Classifying homogeneous Riemannian manifolds

Much harder.

⁸

Footnotes

1. https://mathoverflow.net/questions/346364/is-every-homogeneous-space-riemannian-homogeneous ↩

2. https://math.stackexchange.com/a/3876991/1290493 ↩

3. dg.differential geometry - How do you see that higher genus surfaces are not homogeneous? - MathOverflow ↩

4. differential geometry - Proving that given any two points in a connected manifold, there exists a diffeomorphism taking one to the other - Mathematics Stack Exchange ↩

5. dg.differential geometry - Example of a manifold which is not a homogeneous space of any Lie group - MathOverflow ↩

6. A Structure Theorem for Homogeneous Spaces | Geometriae Dedicata ↩

7. https://mathoverflow.net/a/104106 ↩

8. https://math.stackexchange.com/a/4920625 ↩