Preserving problems of geodesic-affine maps and related topics on positive definite matrices

Based on affine maps in geometry, we study the geodesic-affine maps on Riemannian manifolds PnPn of complex positive definite matrices that are induced by different so-called kernel functions. In this article, we are going to describe the structure of all continuous bijective geodesic-affine maps on these manifolds. We also prove 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.