I'm studying properties of the manifold of symmetric positive-definite (SPD) matrices and I've learnt about the following connection to the hyperbolic space [1, Section 2.2], $$\mathcal{P}(n) = \mathcal{SP}(n) \times \mathbb{R}^+,\qquad \mathcal{SP}(n) \cong \mathbb{H}^p,$$ where $\mathcal{P}(n)$ is the space of SPD matrices, $\mathcal{SP}(n) = \{ A \in \mathcal{P}(n) : \lvert A \rvert = 1 \}$, and $p = n (n + 1) / 2 - 1$. In words, 



> 
> $\mathcal{P}(n)$ is a foliated manifold whose codimension-one leaves are isomorphic to the hyperbolic space $\mathbb{H}^p$.
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I wanted to see a proof of this isomorphism (and what it means more precisely) and I found [2] and its follow up [3] which focus on $2 \times 2$ SPD matrices. They mention certain connections between the standard metrics in these spaces, e.g. [3, p4],
\begin{align}\tag{1}\label{eq:1}
d\_{\mathcal{P}}(X\_1,X\_2) = \sqrt{\frac{1}{2} \big(\log \lvert X\_1 \rvert - \log \lvert X\_2 \rvert \big)^2 + d\_{\mathbb{D}}^2(y\_1,y\_2)},
\end{align}
where $d\_{\mathcal{P}}$ is the canonical metric on $\mathcal{P}(n)$ (see [2, eq. (6)]), $d\_{\mathbb{D}}(y\_1,y\_2)$ is the distance in the Poincaré disk between two points that "correspond" to the SPD matrices $X\_1, X\_2$ (more precisely, their scaled versions $\tilde{X}\_1,\tilde{X}\_2 \in \mathcal{SP}(n)$; see [2, p4-6]).


That being said, I couldn't find a proof for the general case of $n \times n$ SPD matrices. Is anyone aware of other relevant resources? Or is it really obvious? Does a generalized form of \eqref{eq:1} still hold?


Thank you.




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[1]: Moakher, M. (2005). A differential geometric approach to the geometric mean of symmetric positive-definite matrices. SIAM Journal on Matrix Analysis and Applications, 26(3), 735-747.


[2]: Chossat, P., & Faugeras, O. (2009). Hyperbolic planforms in relation to visual edges and textures perception. PLoS Computational Biology, 5(12), e1000625.


[3]: Faye, G., Chossat, P., & Faugeras, O. (2011). Analysis of a hyperbolic geometric model for visual texture perception. The Journal of Mathematical Neuroscience, 1(1), 4.



Isomorphism between the hyperbolic space and the manifold of SPD matrices with constant determinant
This 'isomorphism' does not hold for $n>2$, in the sense that the 'natural' $\mathrm{SL}(n,\mathbb{R})$-invariant metric on what you are calling $\mathcal{SP}(n)$ and the constant sectional curvature hyperbolic metric on $\mathbb{H}^p$ for $p = n(n{+}1)/2-1$ are only isometric when $n=1$ (the trivial case) and $n=2$. 


The reason is that $\mathcal{SP}(n) = \mathrm{SL}(n,\mathbb{R})/\mathrm{SO}(n)$ as symmetric spaces while $\mathbb{H}^p = \mathrm{SO}(p,1)/\mathrm{SO}(p)$ as symmetric spaces. In fact, for $n>2$, the symmetric space $\mathrm{SL}(n,\mathbb{R})/\mathrm{SO}(n)$ does not have constant sectional curvature, while hyperbolic space $\mathrm{SO}(p,1)/\mathrm{SO}(p)$ has constant sectional curvature for all $p$. 


When $n=2$, it's one of the 'accidental' isomorphisms that $\mathrm{SO}(2,1)$ is double-covered by $\mathrm{SL}(2,\mathbb{R})$. For $p>2$, the group $\mathrm{SO}(p,1)$ is not evenly-covered by any $\mathrm{SL}(n,\mathbb{R})$ for any $n$.


