I am interested in the sectional curvatures of the manifold of symmetric positive definite $n \times n$ matrices with the affine metric and more precisely in a tight lower bound. It's fairly well known that this manifold has non-positive sectional curvatures, but I cannot find an exact expression in the literature. The best I am able to find is an expression for the scalar curvature in this paper https://arxiv.org/pdf/1807.01113.pdf (end of page 8), which should imply an expression for the average of sectional curvatures at each point and, concequently, a rough lower bound for them. However, is there any tighter expression for the sectional curvatures of this manifold (or at least for a lower bound)?
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$\begingroup$ I don't know what is the affine metric, but I have a feeling that describing it as a homogeneous space may help. $\endgroup$– MalkounCommented Nov 11, 2020 at 17:57
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Dolcetti and Pertici give an explicit expression for the Riemannian curvature tensor on the symmetric space $\mathcal{P}_n$ of $n \times n$ positive definite matrices with affine-invariant metric. Using this expression, Criscitiello and Boumal (Prop. I.1 in Appendix I) show that the sectional curvatures of $\mathcal{P}_n$ are contained in interval $[-\frac{1}{2}, 0]$, and this bound is tight.
