Semi-simple isometries of the manifold of symmetric positive definite real matrices with the trace metric

We study the semi-simple (or non-parabolic) isometries Φ Φ of the manifold Pₙ P n of symmetric positive definite real matrices, endowed with the trace metric g and pay attention to the set Min(Φ ) M i n ( Φ ) consisting of the points minimizing the displacement function P ↦ d(P, Φ (P)) P ↦ d ( P , Φ ( P ) ) . In particular, for every isometry Φ Φ of (Pₙ, g) ( P n , g ) , we give the explicit expression and study the differential-geometric structure of the set Min(Φ ) M i n ( Φ ) .