# (Locally) sym., homogenous spaces and space forms

We had some definitions of particular types of Riemannian manifolds in our lecture

1.) Locally symmetric spaces. They were Riemannian manifolds with the property that $\nabla R=0$ everywhere.

2.) Symmetric spaces. They were path-conn. Riemannian manifolds such that there is for each $p \in M$ a global isometry $f_p: M \rightarrow M$ such that $f(\gamma(t)) = f(\gamma(-t))$ for all geodesics $\gamma: (-\varepsilon,\varepsilon) \rightarrow M$ satisfying $\gamma(0)=p$ and $Df_p(p) = -id.$

3.) Homogenous spaces. They were maps admitting for every $p,q\in M$ a global isometry $\phi_{p,q}: M \rightarrow M$ such that $\phi(p)=q$

4.) Finally, we introduced space-forms as complete connected Riemannian manifolds of const. curvature.

Now, to enhance my understanding of all these spaces, I wanted to get the relations between these spaces right.

So, I found out that any symm. space is homogenous and every symm. space is locally symmetric. Furthermore, any symm. space is complete and path.-connected so i.e. a space form.

Now, I have basically three questions that I could not really answer

1.) Is there a way to compare homogenous, loc. symm. spaces and space forms, too?

2.) Which of these spaces have constant sectional curvature everywhere? By the global isometry property I guess it holds for hom., symm. spaces and space forms, but I don't know if this is also true for loc. symm. spaces?

3.) Are there easy(!) examples of spaces that are one but not the other? ( Naming them woud be totally sufficient, I would try to figure it out by myself why they are examples)

• This is a monstrously huge question. Basically you are asking for an entire course in one stackexchange answer. You might get a lot more mileage by breaking this into one small answerable question at a time. – Lee Mosher Jun 20 '15 at 16:10
• @LeeMosher you think I should ask all three questions separately? Actually I refrained from doing so, cause I thought that $1$ can be answered in one sentence, as either you can say that one of them is a subset of another one or not. $2$ is basically a yes, no question, because either loc. symm. spaces have constant sectional curvature or not and in $3$ I asked only for the examples (without any details). Furthermore, all questions are all highly related but if you still think that a splitting is appropriate, please leave me a comment and I will do it. – wewasss Jun 20 '15 at 16:17

You need examples. The archetypal example of locally symmetric but non homogeneous space is a compact quotient of the unit disk. Namely, a compact Riemannian surface with its canonical metric also called hyperbolic Riemann surfaces. Here you have more details: https://en.wikipedia.org/wiki/Riemann_surface

So usually the difference between locally symmetric and symmetric is related to the topology or the lack of completeness.

The archetypal examples of homogeneous but not locally symmetric space are given by a Lie group endowed by a so called left invariant metric (some of them are of course symmetric but in general they are not). Between them my favorite one is the Heisenberg group https://en.wikipedia.org/wiki/Heisenberg_group endowed with a left invariant metric.

The archetypal example of symmetric space that is not a space form is the complex plane $\mathbb{C}P^2$ endowed with its Fubini-Study metric.

• As I wrote the complex projective $\mathbb{C}P^2$ endowed with the Fubini-Study metric is a symmetric space but has not constant sectional curvature: en.wikipedia.org/wiki/Fubini%E2%80%93Study_metric . As far as I see this answer question (2), in the sense that show you the existence of a homogeneous spaces, even symmetric whose sectional curvature is not constant. – Holonomia Jun 20 '15 at 18:26