# Manifolds and Lorentz-group

1. Jan 10, 2009

### parton

1. The problem statement, all variables and given/known data
I've got a problem. I should discribe all minimal invariant manifolds in Minkowski-space, where the proper Lorentz-group $$\mathcal{L}_{+}^{\uparrow}$$ acts transitivly (i.e. any two points of the manifold can be transformed into each other by a Lorentz-transformation).

2. Relevant equations

3. The attempt at a solution
My problem is that I don't really understand what I should do. For example: what is meant by a minimal invariant manifold? What has a manifold to do with a group at all?

The only thing that I know for sure is: $$\Lambda \in \mathcal{L}_{+}^{\uparrow} \Rightarrow \mathrm{det} \Lambda = 1$$ and $$\Lambda^{0}_{0} \geq 1$$ and $$\mathcal{L}_{+}^{\uparrow}$$ contains rotations and Lorentz-boosts.

I'm really looking forward to get some help for this exercise.