I start by stating what I think I understood on symmetric cones (https://en.wikipedia.org/wiki/Symmetric_cone). Let $\mathcal{C}$ be a symmetric cone in a vector space $V$. There are Riemannian metrics on $\mathcal{C}$, invariant by the elements of $GL(V)$ which fix $\mathcal{C}$. Let $G$ be such a metric, $(\mathcal{C},G)$ is then a Riemannian symmetric space.
Let $\mathcal{S}= \mathcal{C}/ \mathbb{R}_{>0}$ be the manifold of lines of the cone. I have in mind that
- $G$ descends on $\mathcal{S}$ and gives it a structure of Riemannian symmetric space of non-compact type (maybe under some irreducibility assumption)
- with this metric on $\mathcal{S}$, $\mathcal{C}$ is isometric to a product $\mathcal{S}\times \mathbb{R}_{>0}$, where $\mathbb{R}_{>0}$ is endowed with the logarithmic metric.
Is that correct ? I have trouble proving it on my own or finding reference for this. Thank you for your help.