Abstract:   Based on affine maps in geometry, we study the geodesic-affine maps on Riemannian manifolds P_n of complex positive definite matrices that are induced by different so-called kernel functions. In this talk, we are going to describe the structure of all continuous bijective  geodesic-affine maps on these manifolds. We also show that geodesic distance isometries are geodesic-affine maps.  Moreover, the forms of all bijective maps which preserve norms of geodesic correspondence are characterized. Indeed, these maps are special examples of geodesic-affine maps.