Let $G/H$ be an affine symmetric space with involution $\sigma$, and $\mathfrak{g}=\mathfrak{m}\oplus \mathfrak{h}$ the Cartan decomposition of its Lie algebra. We can identify $G/H$ and $\exp(\mathfrak{m})$. Given $p,q\in \exp(\mathfrak{m})$, I am wondering how to compute the holonomy of the curve formed by the geodesic segments, $e\rightarrow p$, $p\rightarrow q$, $q\rightarrow e$, $e$ being the identity. If I understand correctly, the holonomy should be given by a $Ad_k$ with $k\in H$. Can we compute $k$ in closed form from $p$ and $q$? I suspect it might be possible but I didn't manage.

Thank you for your help.


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