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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 $\mathrm{GL}(V)$ 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

  1. $G$ descends on $\mathcal{S}$ and gives it a structure of Riemannian symmetric space of non-compact type (maybe under some irreducibility assumption)
  2. 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.

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    $\begingroup$ Cross-posted at MSE here: You should avoid simultaneous posting at MSE and MO in order to avoid duplication of efforts. $\endgroup$
    – Moishe Kohan
    Commented May 15, 2023 at 12:30
  • $\begingroup$ On substance: (2) is true and (1) (if I remember it correctly) is false even in the Hermitian case, because of an example by Piatetsky-Shapiro. $\endgroup$
    – Moishe Kohan
    Commented May 15, 2023 at 12:32

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